Find the volume of the solid obtained by rotating the region bounded by y=x^2 and x=y^2. Rotating about y=1.

dredyue

dredyue

Answered question

2022-08-11

How can I find the volume of a solid of revolution
Find the volume of the solid obtained by rotating the region bounded by y = x 2 and x = y 2 . Rotating about y = 1.
I got an intercept of those functions which was (1,1). I tried to use washer method then I got
π 0 1 [ ( x ) 2 ( x ) 2 ] d x and I took integral of the functions but my volume was not right number. I think my way to solve was not right. Could you post correct way to find the volume?

Answer & Explanation

Evelin Castillo

Evelin Castillo

Beginner2022-08-12Added 12 answers

Step 1
Your formula assumes that you are rotating about the x-axis (that is, about y = 0), but you aren't; you are rotating about y = 1. Also, you forgot that your other function was y = x 2 and not y = x(the ( x ) 2 factor was probably meant to be ( x 2 ) 2 ?).
If you have a thin strip of width Δ x that starts at the point x i , then you have a washer with outer radius 1 x i 2 and inner radius 1 x i (draw a picture, and remember the axis of rotation). So the volume described by the washer is approximated by π ( ( 1 x i 2 ) 2 ( 1 x i ) 2 ) Δ x .
Step 2
Take the Riemann sum, then limits as Δ x 0 to turn it into an integral.

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