`y = x^2, x = y^2` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. (about y = 1)

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The volume of the solid obtained by rotating the region bounded by the curves `y^2=x` and `y=x^2` about y = 1, can be evaluated using the washer method, such that:

`V = int_a^b pi*(f^2(x) - g^2(x))dx`

You need to find the endpoint of interval, hence, you need to solve for...

Unlock
This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Start your 48-Hour Free Trial

The volume of the solid obtained by rotating the region bounded by the curves `y^2=x` and `y=x^2` about y = 1, can be evaluated using the washer method, such that:

`V = int_a^b pi*(f^2(x) - g^2(x))dx`

You need to find the endpoint of interval, hence, you need to solve for x the following equation, such that:

`sqrt x = x^2 => x - x^4 = 0 => x(1 - x^3) = 0 => x = 0` and `x = 1`

You need to notice that `x^2 < sqrt x` on [0,1], such that:

`V = int_0^1 pi*(((sqrt x) - 1)^2 - (x^2 - 1)^2)dx`

`V = pi*int_0^1 (x- 2sqrt x + 1)dx - pi*int_0^1 (x^4-2x^2 + 1)dx`

`V = (pi*x^2/2 - (4/3)pi*x^(3/2) + pi*x - pi*x^5/5 + 2pi*x^3/3 - pi*x)|_0^1`

`V = pi*1^2/2 - (4/3)pi*1^(3/2) - pi*1^5/5 + 2pi*1^3/3 - 0`

`V = pi/2 + (4pi)/3 - pi/5 - (2pi)/3`

`V = pi/2 + (2pi)/3 - pi/5`

`V = (15pi + 20pi - 6pi)/30`

`V = (29pi)/30`

Hence, evaluating the volume of the solid obtained by rotating the region bounded by the curves `y^2=x , y=x^2` about y = 1 , using the washer method, yields `V = (29pi)/30.` ` `

Approved by eNotes Editorial Team