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AreaFind the area between y=x and f(x)=x^2.

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chezmena | Student, College Freshman | eNoter

Posted May 11, 2011 at 1:53 AM via web

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Area

Find the area between y=x and f(x)=x^2.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted March 7, 2013 at 6:53 PM (Answer #3)

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You need to evaluate the limits of integration, hence, you need to solve the following equation, such that:

`x^2 = x => x^2 - x = 0 => x(x - 1) = 0 => {(x = 0),(x - 1 = 0):}`

`{(x = 0),(x = 1):}`

You need to evaluate the area of the region bounded by the given curves, hence, you need to evaluate the following definite integral, such that:

`A = int_0^1(x - x^2)dx` (` y = x > y = x^2` over `x in [0,1]` )

You need to use the property of linearity of integral, such that:

`A = int_0^1 x dx - int_0^1 x^2 dx`

`A = (x^2/2 - x^3/3)|_0^1`

`A = (1/2 - 0/2 - 1/3 + 0/3) => A = (1/2 - 1/3)`

`A = 1/6`

Hence, evaluating the area of the region bounded by the given curves yields `A = 1/6.`

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giorgiana1976 | College Teacher | Valedictorian

Posted May 11, 2011 at 10:43 AM (Answer #2)

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You've given only the limit curves but you did not gave the limit lines.

We'll calculate the area located between the 2 given curves, using Leibniz-Newton formula:

Int (x^2 - x)dx = F(b) - F(a), where x = a and x = b are the limit lines.

We'll determine the indefinite integral first:

Int (x^2 - x)dx = Int x^2 dx - Int x dx

Int (x^2 - x)dx = x^3/3 - x^2/2

F(b) = b^3/3 - b^2/2

F(a) = a^3/3 - a^2/2

Int (x^2 - x)dx  = b^3/3 - b^2/2 - a^3/3 + a^2/2

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