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Generalizing disc method around x-axis | AP Calculus AB | Khan Academy

Video vocabulary

essentially

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US / ɪˈsenʃəli /

UK / ɪˈsenʃəli /

A2
  • adverb
  • Basically; (said when stating the basic facts)

right

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US / raɪt /

UK / raɪt /

A1
  • adjecitve
  • Correct or true
  • The south side of you looking at the rising sun
  • adverb
  • Exactly in a place or time
  • Being appropriate for a particular event
  • In a straight or direct manner
  • noun
  • Something you are legally or morally allowed to do
  • Ideal of what is just and good
  • Political view based on conservatism
  • verb
  • To fix something previously wrong or unjust
  • To make upright, as a boat that has turned over

disk

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US / dɪsk /

UK / dɪsk /

B1
  • noun
  • Thin or flat, round object, e.g. for storing data

depth

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US / depθ /

UK / dɛpθ /

A2
  • noun
  • Distance below a surface

find

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US / faɪnd /

UK / faɪnd /

A1
  • noun
  • Object discovered by chance
  • Things that have been discovered from searching
  • verb
  • To become aware of something that is happening
  • To discover or meet by chance
  • To decide if someone is guilty of a crime, or not
  • To learn through study, research, or effort
  • To discover something by looking for it

first

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US / fɜ:st /

UK / fɚst /

A1
  • adjecitve
  • 1st; Coming before all others in time or place
  • noun
  • Something that has not happened before
  • Position of the winner of a competition

disc

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US / dɪsk /

UK / dɪsk /

A2
  • noun
  • Thin or flat, round object, e.g. for storing data

get

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US / ɡet /

UK / ɡɛt /

A1
  • verb
  • To become affected by illness or disease
  • To begin to feel or understand an emotion or idea
  • To send or bring someone or something home, etc.
  • To go somewhere to obtain something
  • To (cause to) do a particular thing
  • To obtain, receive or be given something
  • To currently have
  • To prepare for use
  • To understand something being said or read

number

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US / ˈnʌmbə(r) /

UK / ˈnʌmbɚ /

A1
  • noun
  • Symbols such as 1, 2, 56, 793
  • Particular song or dance performed during a show
  • Total quantity or amount of things
  • verb
  • To put numbers on things
  • To assign a sequence within a group, series, set
  • To claim to be part of a larger group, series, set

square

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US / skweə(r) /

UK / skwɛr /

A2
  • adjecitve
  • Being fair or honest
  • Multiply a number by itself e.g. 2x2 or 6x6
  • noun
  • Shape with 4 equal length sides and 4 equal angles
  • The result of multiplying Length by the width
  • Open public area in the middle of the town
  • verb
  • To make something balance or be equal
  • To multiply a number by itself, e.g. 2x2 or 6x8
  • To straighten the edge to form a four-sided shape

like

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US / laɪk /

UK / laɪk /

A1
  • adjecitve
  • Being equal in status or kind to something else
  • Similar; the same
  • adverb
  • For example
  • preposition
  • Used as a filler; used to ad emphasis
  • In a way similar to the way something is done
  • Such as; for example
  • verb
  • To want to
  • To find something pleasing; to prefer something

what

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US / wɒt /

UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /

A1
  • pronoun
  • Thing or things that; the same kind as
  • Used to emphasize something you are about to say
  • adjecitve
  • Used to refer to a particular amount, or number
  • Used to place emphasis on a particular quality
  • adverb
  • To a certain degree
  • determiner
  • Question words used for asking for information
  • exclamation
  • Used to express excitement, shock, or surprise
  • Used to ask people to repeat something they said

that

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US / ðæt /

UK / ðæt, ðət /

A1
  • adjecitve
  • The person or thing being talked about
  • Used to identify something both the speakers know
  • adverb
  • Used to reinforce adjectives and adverbs
  • To such a degree
  • conjunction
  • Used to connect a noun clause
  • Used to introduce an adverbial clause
  • Used to introduce a noun clause
  • determiner
  • Used as a noun to refer to something
  • pronoun
  • Referring an object far away
  • Used to refer to the relative pronoun ‘which’

each

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US / i:tʃ /

UK / itʃ /

A1
  • determiner
  • Every one of two or more things
  • pronoun
  • Per one person or thing

method

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US / 'meθəd /

UK / ˈmɛθəd /

A2
  • noun
  • (Organized and planned) way of doing something

recommend

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US / ˌrekə'mend /

UK / ˌrɛkəˈmɛnd /

A2
  • verb
  • To advise or suggest that someone do something

then

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US / ðen /

UK / ðɛn /

A1
  • adverb
  • A time that is not now
  • Following another thing in time or arrangement
  • Following on reasonably from what was just stated
  • At that time, not now

advise

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US / ədˈvaɪz /

UK / ædˈvaɪz /

A2
  • verb
  • To suggest what would be helpful

know

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US / nəʊ /

UK / noʊ /

A1
  • verb
  • To be familiar with a person or place, thing
  • To feel sure or be convinced about something
  • To discover or be aware of something
  • To be able to distinguish
  • To have knowledge of things

often

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US / ˈɔfən /

UK / ˈɔfən,ˈɑfən,ˈɔftən,ˈɑf- /

A1
  • adverb
  • Many times; frequently; on many occasions

see

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US / si: /

UK / si /

A1
  • verb
  • To go on a date with someone
  • To predict whether something is, or will be true
  • To understand what someone says
  • To use your eyes to look at something
  • To visit a place often for sightseeing or pleasure
  • To visit or meet with someone
  • To find out by waiting or looking
  • To watch (a game, movie or TV show)

end

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US / end /

UK / ɛnd /

A1
  • noun
  • The point where a process or activity stops
  • A point that marks the limit of something; finish
  • Goal or desired result
  • The tip, edge or limit of something
  • verb
  • To be the last part of something

have

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US / həv /

UK / hæv /

A1
  • verb
  • To drink, smoke, eat or use something
  • To experience the effects of something
  • To organize an event, or join in an activity
  • To own, possess, or hold something
  • To cause to happen or produce a particular effect

last

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US / lɑ:st /

UK / læst /

A1
  • adjecitve
  • Most recent; the latest
  • The one before
  • adverb
  • Final; with nothing following; after all others
  • Most recent; the latest
  • determiner
  • The one at the end
  • verb
  • To remain or continue for a certain period of time

part

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US / pɑ:t /

UK / pɑ:rt /

A1
  • adjecitve
  • To some degree; half; not fully
  • noun
  • Division of a book
  • Ratio of something, e.g. 3 of gin, 1 of tonic
  • A line in a person's hair, made with a comb
  • Character an actor plays on stage, movie, or TV
  • Some, but not all of a specific thing
  • Role in causing something to happen
  • verb
  • To make a line in a person's hair, by using a comb
  • To move two things apart creating a space
  • To leave someone, or to break off a relationship

with

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US / wɪð /

UK / wɪð, wɪθ /

A1
  • preposition
  • As an employee of a certain company
  • Against something or someone
  • By using something as a means
  • Due to; because of
  • Including
  • Possessing as a feature
  • Related to
  • (Of the manner/attitude by which you do something)
  • Used to express that people or things are together

let

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US / let /

UK / lɛt /

A1
  • verb
  • To allow someone to do something
  • Introducing a suggestion to do something together
  • To rent a house, etc. to others

the

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US / ðə /

UK / ðə /

A1
  • article
  • Used to refer to something already mentioned
  • Used to show there is only one of something
  • determiner
  • Used to describe all of a family

this

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US / ðɪs /

UK / ðɪs /

A1
  • adjecitve
  • Used to indicate something already been discussed
  • Used to indicate something happening around now
  • adverb
  • An indication by a physical motion or gesture
  • A specific amount of
  • determiner
  • Person, thing, or idea near you
  • pronoun
  • Person, thing that has already been discussed

how

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US / haʊ /

UK / haʊ /

A1
  • adjecitve
  • To what degree? in what amount?
  • adverb
  • (Used to explain the way to do something)
  • exclamation
  • For what purpose or reason?
  • (Used to ask someone or something's condition)

simply

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US / ˈsɪmpli /

UK / ˈsɪmpli /

A1
  • adverb
  • In an easy or clear manner

you

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US / jʊ /

UK / ju /

A1
  • pronoun
  • Person someone is speaking or writing to
  • Person or people in general

show

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US / ʃəʊ /

UK / ʃo /

A1
  • noun
  • False display of love, emotion or action
  • An event for displaying or promoting goods
  • Play, opera or theater production
  • Presentation that is broadcast on TV or radio
  • verb
  • To be easily seen or displayed
  • To display your emotions or feelings
  • To let someone see something; display
  • To explain or teach how something is done
  • To lead someone somewhere; point them to something
  • To prove something to be true, correct
  • To reveal or allow something to be seen

space

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US / speɪs /

UK / spes /

A1
  • noun
  • Empty area kept for a specific reason, like a car
  • Empty area with nothing in it
  • Area in the sky where the stars are
  • Area between written words on a page
  • verb
  • To make a certain distance or time between

multiply

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US / ˈmʌltɪplaɪ /

UK / ˈmʌltəˌplaɪ /

B1
  • verb
  • To increase in number by reproducing/having babies
  • To increase something by a large quantity
  • To have increased by a large quantity
  • To calculate two numbers e.g. 2 x 4

all

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US / ɔ:l /

UK / ɔl /

A1
  • adverb
  • Completely; totally
  • determiner
  • Being every one of something
  • pronoun
  • 100% of something; the complete amount

common

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US / 'kɒmən /

UK / ˈkɑmən /

A1
  • adjecitve
  • Shared; Belonging to or used by everyone
  • Typical, normal; not unusual
  • noun
  • Area in a city or town that is open to everyone
  • Field near a village owned by the local community

between

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US / bɪˈtwi:n /

UK / bɪˈtwin /

A1
  • preposition
  • From one person, thing, or place, to another
  • In the space that separates two objects
  • Among two or more people who share something

principle

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US / ˈprɪnsəpl /

UK / ˈprɪnsəpəl /

A2
  • noun
  • Moral rule or belief governing a person's behavior
  • Natural law that explains how things work

they

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US / ðeɪ /

UK / ðe /

A1
  • pronoun
  • Two or more people, animals, or things

volume

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US / ˈvɒlju:m /

UK / ˈvɑljum, -jəm /

A2
  • noun
  • Number or amount of something e.g. sales
  • Amount of space that something takes up
  • One book that is part of a set of books
  • Level of sound produced by radio or television

take

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US / teɪk /

UK / tek /

A1
  • noun
  • Filming of a single scene for TV or the movies
  • verb
  • To accept something that is offered
  • To use a method of traveling, e.g. a bus or train
  • To use a camera to make (a photo)
  • To pick up something and go away with it
  • To study a subject in school
  • To attempt to pass (an exam)
  • To require a certain amount of time, money, space

actually

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US / ˈæktʃuəli /

UK / ˈæktʃuəli /

A1
  • adverb
  • Used to add new (often different) information

integral

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US / ˈɪntɪgrəl /

UK / ˈɪntɪɡrəl, ɪnˈtɛɡrəl /

B1
  • adjecitve
  • Being an essential part of something; necessary

use

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US / ju:s /

UK / ju:s /

A1
  • noun
  • Ability to employ something
  • Need to enable it to work
  • Permission to do something with something
  • Act of spending money
  • Operation of a machine for its function
  • verb
  • To have; to eat, drink, consume etc.
  • To need to operate or work
  • To operate a machine, e.g. a computer
  • To treat a person badly
  • To do something with, for a task or purpose

area

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US / 'eərɪə /

UK / ˈɛriə /

A1
  • noun
  • Amount of measured space
  • A specific section or space; part of a region
  • A subject or field of study

will

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US / wɪl /

UK / wɪl /

A1
  • noun
  • Desire or choice of someone
  • Document saying who gets your money when you die
  • Strong desire to do something; determination
  • Desire to do something; strong wish to do
  • other
  • Used with verbs to express the future
  • verb
  • To leave property to someone after your death
  • To use mental effort to make something happen
  • Am (is, are) likely to do
  • To influence someone to do something
  • To make something happen by strongly wanting it

inside

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US / ˌɪn'saɪd /

UK / ɪnˈsaɪd, ˈɪnˌsaɪd /

A2
  • adjecitve
  • Being in something
  • Being in a building
  • adverb
  • Into a house; indoors
  • noun
  • Inner part or surface of something

here

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US / hɪə(r) /

UK / hɪr /

A1
  • adverb
  • In this place; at this location, position or time

small

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US / smɔ:l /

UK / smɔl /

A1
  • adjecitve
  • Little in size; not big
  • (Of letters) not written in capital letters
  • Being insignificant or minor
  • (Of a business) with few people or things; local
  • (Of voice) being soft and quiet
  • (Of a child) being young

point

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US / pɔɪnt /

UK / pɔɪnt /

A1
  • noun
  • An item to be discussed
  • Small spot or dot
  • Fact or detail
  • Feature of something
  • The size of the text on a page or screen
  • A certain time or stage in a process
  • Certain position in time or space
  • Idea or fact you try to convince people about
  • Measurement of scores in a game, sport
  • Sharp end of something, e.g. a pen or pin
  • verb
  • To face a certain direction, e.g. north
  • To indicate something with your finger to others

really

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US / ˈri:əli /

UK / ˈriəˌli, ˈrili /

A1
  • adverb
  • Used to stress something that is said; definitely
  • In a true or accurate manner
  • Very

would

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US / wʊd /

UK / wʊd /

A1
  • other
  • Used to indicate something imagined
  • Used to ask or request something politely
  • Verb used to indicate possibility, request etc.
  • Used to indicate a preference
  • Used to indicate a reaction
  • Indicates something that occurred repeatedly
  • Used to indicate willingness or ability to do

sense

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US / sens /

UK / sɛns /

A1
  • noun
  • Certain mental feeling or emotion
  • Normal or clear state of mind
  • Meaning of a particular word, phrase or text
  • verb
  • To perceive using sight, sound, taste touch etc.
  • To recognize the presence of something

sum

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US / sʌm /

UK / sʌm /

A2
  • noun
  • Amount when all is added together; total
  • verb
  • To add two or more numbers together; total

video

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US / 'vɪdɪəʊ /

UK / ˈvɪdiˌo /

A2
  • adjecitve
  • Of a moving film, with sound
  • noun
  • Recording capturing action with sound
  • verb
  • To make a moving film of, with sound

want

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US / wɔnt /

UK / wɑnt, wɔnt /

A1
  • noun
  • Something you desire or wish for
  • State or condition of poverty
  • verb
  • To desire or wish for something; hope for a thing

same

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US / seɪm /

UK / sem /

A1
  • adjecitve
  • Being equal in quality and amount
  • Identical; not different
  • Not changing; constant

but

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US / bət /

UK / bʌt,bət /

A1
  • conjunction
  • Used before you say something different, opposite

our

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US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

not

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US / nɒt /

UK / nɑt /

A1
  • adverb
  • Word indicating the negative
  • (Used to form the negative of verbs)

textbook

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US / ˈtekstbʊk /

UK / ˈtɛkstˌbʊk /

A1
  • noun
  • A book that is used to study

against

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US / əˈgenst /

UK / əˈɡenst /

A1
  • preposition
  • In physical contact with, as for support; next to
  • In opposition to; disagreeing with

formula

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US / 'fɔ:mjələ /

UK / ˈfɔrmjələ /

A2
  • noun
  • A mathematical or scientific rule
  • Plan, rule or method for doing or making something

radius

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US / ˈreɪdiəs /

UK / ˈrediəs /

B2
  • noun
  • Area around a certain point
  • Long bone on the thumb side of your lower arm
  • Line from the center to the edge of a circle

something

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US / ˈsʌmθɪŋ /

UK / ˈsʌmθɪŋ /

A1
  • adverb
  • In some degree or measure
  • noun
  • a person or thing of some value or consequence
  • pronoun
  • Thing that is not yet known or named
  • Approximately

come

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US / kʌm /

UK / kʌm /

A1
  • verb
  • To arrive at a place
  • (Of mail) to be delivered
  • To move toward someone; go with someone
  • To reach a certain point or state

call

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US / kɔ:l /

UK / kɔl /

A1
  • noun
  • A order or request for action
  • The sound an animal makes, often when in danger
  • A correct prediction, e.g. a decision in a game
  • A short visit to a place or person
  • A decision by a referee or umpire in a game
  • verb
  • To make a request or order for action
  • (Animal) to make sounds to warn of danger
  • To announce the date of an election
  • To make a judgment about something
  • To give someone or something a name
  • To guess in advance the result or outcome
  • To visit a place or person for a short time
  • To phone someone
  • To speak, announce or shout in a loud voice

where

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US / hwɛə /

UK / hwɛr, wɛr /

A1
  • conjunction
  • At or in the place which
  • In whatever place
  • pronoun
  • Word you use to ask the location of something

along

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US / əˈlɒŋ /

UK / əˈlɔ:ŋ /

A1
  • adverb
  • As a companion; taken with someone
  • preposition
  • At a point on a line

from

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US / frəm /

UK / frʌm, frɑm,frəm /

A1
  • preposition
  • Starting at a particular place, time or level

infinite

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US / 'ɪnfɪnət /

UK / ˈɪnfənɪt /

B2
  • adjecitve
  • Being very large in amount
  • Without an end; limitless; endless

out

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US / aʊt /

UK / aʊt /

A1
  • adverb
  • No longer stylish (e.g. fashion)
  • Away from home, business, or the usual place
  • In a direction/movement away from inside or center
  • noun
  • Act when a batter cannot continue playing
  • verb
  • To tell people a secret

over

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US / 'əʊvə(r) /

UK / ˈovɚ /

A1
  • adverb
  • Finished; no longer happening
  • Towards the ground/away from a standing position
  • noun
  • A set of six consecutive balls bowled in cricket
  • other
  • Finished
  • preposition
  • Showing the relation when a thing is above another
  • On or covering something
  • Above or across something
  • More than you need
  • From one side to the other
  • In the direction of; toward

height

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US / haɪt /

UK / haɪt /

A2
  • noun
  • Distance of something from the bottom to the top

two

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US / tu: /

UK / tu /

A1
  • number
  • 2

and

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US / ənd /

UK / ənd, ən,ænd /

A1
  • conjunction
  • Used to refer to two or more things
  • Plus; in addition; on top of that
  • Used to introduce an action that follows another

just

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US / dʒʌst /

UK / dʒʌst /

A1
  • adjecitve
  • Proper or appropriate; as is deserved
  • Doing or being what is right or fair
  • Legally correct; having enough evidence
  • adverb
  • Almost; nearly
  • Exactly at the time of
  • Exactly (the same as, like)

graph

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US / græf /

UK / ɡræf /

A2
  • noun
  • Image expressing numerical data in lines or shapes
  • verb
  • To show data using images

highly

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US / ˈhaɪli /

UK / 'haɪlɪ /

A1
  • adverb
  • In a manner that shows approval
  • More than usual; to a great degree

memorize

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US / ˈmeməraɪz /

UK / ˈmɛməˌraɪz /

A2
  • verb
  • To study something so that you remember it

for

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US / fə(r) /

UK / fɔr,fə /

A1
  • preposition
  • Used to show the purpose, or need of something

time

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US / taɪm /

UK / taɪm /

A1
  • noun
  • Speed at which music is played; tempo
  • Point as shown on a clock, e.g. 3 p.m
  • Number of hours, minutes needed to do something
  • Occasion when something happens
  • Period or occasion that something occurred
  • Period in history or the past
  • Something measured in minutes, hours, days, etc.
  • How long an event takes; duration
  • verb
  • To check speed at which music is performed
  • To choose a specific moment to do something
  • To measure how long an event takes, e.g. a race
  • To schedule something to occur at a specific moment

face

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US / feɪs /

UK / fes /

A1
  • noun
  • Side or outer surface of something
  • An expression using mouth, nose, eyes etc.
  • Front part of the head where eyes, mouth etc. are
  • Characteristics you show in public
  • verb
  • To cover a surface with something like paint
  • To deal with a problem or responsibilities
  • To look toward; point in a certain direction

exact

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US / ɪɡ'zækt /

UK / ɪɡˈzækt /

A2
  • adjecitve
  • Completely correct; accurate; specific
  • verb
  • To get something, sometimes using force

limit

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US / 'lɪmɪt /

UK / ˈlɪmɪt /

A2
  • noun
  • Point beyond which it is not possible to go
  • verb
  • To stop or prevent an increase past a point

since

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US / sɪns /

UK / sɪns /

A1
  • adverb
  • At a point before the present time
  • conjunction
  • For this reason; because
  • preposition
  • From the time in the past that

well

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US / wel /

UK / wɛl /

A1
  • adjecitve
  • Being in a fortunate, good condition or state
  • Healthy and in good condition
  • adverb
  • In an acceptable, proper, or successful manner
  • In a complete, total, or full manner
  • A great and considerable degree
  • In a successful way, in a good or satisfactory way
  • exclamation
  • Used to begin or continue a particular statement
  • Used to show you want to start saying something
  • Used to express that you are waiting for a reply
  • You say this to change the topic of a conversation
  • Used to indicate you are making a final remark
  • noun
  • Long deep hole in the ground to get water
  • verb
  • To cause your eyes to fill with tears

endpoint

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US / 'endpɔɪt /

UK / 'endˌpɔɪnt /

A1
  • noun
  • The final stage of a period or process.
  • Subtitles section
  • What we're going to do in this video is generalize what we did in the last video.
  • And, essentially, end up with the formula for rotating something around the X-axis like this using, what we call, the "disk method".
  • And the point is to show you where that formula inside a calculus textbook actually comes from.
  • But it just comes from the same exact principles we did in the last video!
  • It's not advised to memorize the formula; I highly recommend against that because you really need to know what's going on.
  • It's really better to do it in first principles where you find the volume of each of these disks.
  • But let's just generalize what we saw in the last video.
  • So instead of saying that "y = x^2", let's just say that this is the graph, the function that's right over here.
  • Let's just generalize it and call it "y=f(x)".
  • And instead of saying x is going from 0 to 2 let's say that we're going between a and b, so these are just two endpoints along the X-axis.
  • So how would we find the volume of this?
  • Well, just like the last video, we still take a disk just like this.
  • But what is the height of the disk?
  • The height of the disk is not just x^2 since we've generalized it.
  • So the height is simply going to be whatever the height of the function is at that point.
  • So the height of the disk is going to be f(x)!
  • The area of the space of this disk is going to be πR^2.
  • So our radius is f(x), and we're just going to square it.
  • That's the area, that's the area of this face right over here.
  • What is the volume of the disk?
  • We're just going to multiply that by our depth, which going to be dx.
  • And we want to take the sum of all of these disks from a to b, and we're going to take the sum of them, and we're going to take the limit as the "dx"s get smaller and smaller.
  • And we have an infinite number of these disks.
  • Thus, we are going to the integral of this from a to b!
  • And this right here is the formula that you will see often in a calculus textbook for using the disk method as you're rotating around the X-axis.
  • So I just wanted to show you that it comes out of the common sense of finding the volume of this disk.
  • The f(x) right over here is just the radius of the disk, so this part over here is really just πR^2; we multiply it [area of the circle] times the depth; then we take the sum from a to b of all of the disks.
  • And essentially since this is integral is the limit of all the disks getting narrower and narrower. And thus, we have an infinite number of these disks.