Find the volume of the region bounded by y=x and y=x^2, but rotated about the equation of y=x.

Gunsaz 2022-09-07 Answered
Volume of Solid of Revolution about an equation
I learned about the disk and shell method for finding a volume of a solid of revolution. For my question I don't think I can use either method directly. Here is my question:
Find the volume of the region bounded by y = x and y = x 2 , but rotated about the equation of y = x.
Here the axis of revolution is not a vertical or horizontal line, but rather the equation y = x. One idea I had was to convert this diagonal line into a horizontal or vertical one, but I can't see to do this. Maybe polar coordinates might be useful, but I'm not sure.
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Answers (2)

emarisidie6
Answered 2022-09-08 Author has 7 answers
Step 1
Another approach using a ''modified'' disk method.
The distance of a point ( x , y ) = ( x , x 2 ) from the line y = x is:
r = | x x 2 | 2
so the area of a circle orthogonal to the axis of rotation is
A = π r 2 = π ( x x 2 ) 2 2
Step 2
And the volume of a disk of length δ x = 2 d x along the axis of rotation is
d V = π 2 0 1 ( x x 2 ) 2 δ x = π 2 0 1 ( x x 2 ) 2 d x
And the volume of the solid of revolution is the integral:
V = π 2 0 1 ( x x 2 ) 2 d x
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rialsv
Answered 2022-09-09 Author has 3 answers
Step 1
Start by writing the curve in parametric form, so x = t , y = t 2 . Now use a matrix to rotate the curve by 45 degrees clockwise so that the new curve is also given parametrically.
( x y ) = 1 2 ( 1 1 1 1 ) ( t t 2 )
Step 2
Now consider the volume as
π t = 0 1 y 2 d x d t d t
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