Homework Help

Find the area of the region: Above y=x^2 and to the right of x=y^2

user profile pic

svjr | Student, Undergraduate | Honors

Posted November 6, 2011 at 4:24 AM via web

dislike 0 like

Find the area of the region: Above y=x^2 and to the right of x=y^2

Tagged with calculus, math

1 Answer | Add Yours

user profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted November 6, 2011 at 5:53 AM (Answer #1)

dislike 0 like

`y= x^2`  (Black curve)

`x = y^2`  ( Red curve)

We need to find the area between both curves.

First we will find the intersection points.

==> `y= x^2 ==gt y= +-sqrt(x).`

`` ==> `x^2 = +-sqrtx`

`` ==> `x^4 = x` ==> `x^4 - x =`  0

==> `x(x^3 -1) =`  0

==> `x (x-1)(x^2 +x +1) = ` 0

==> x = 0

==> x = 1

Then we need to find the bounded area between `x^2`  and `sqrtx`  from 0 to 1.

==> `int_0^1 (x2 -sqrtx) dx = int_0^1 (x^2 - x^(1/2) )`  dx

==> `int_0^1 (x^2 -x^(1/2)) dx = x^3/3 - x^(3/2)/(3/2) = (1/3)x^3 - (2/3)x^(3/2)`

`` ==> (`1^3 - (2/3)(1^(3/2)) = 1- 2/3 = 1/3`

`` ==> `0^3 - (2/3)0^(3/2) =`  0

==> Then, the area is `1/3.`

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes