Finding Volume of the solid

Question

Find the volume of the solid obtained by rotating the region bounded by 𝑦=𝑥^2 , 𝑦=0, and 𝑥=3, 

about the 𝑦-axis.

𝑉= 

Please show work! :D

Answer 1

If you feel that you're unsure on shell method, you can also solve for x and use the washer method to get the same result. In this case, x=√y, so your inner radius is √y and your outer radius is 3 and the bounds would be on y from 0 to 9. So the integral will look like this:

π∫₀⁹ 3² - (√y)² dy --Integrate π times the outer radius squared minus the inner radius squared--

π∫₀⁹ 9 - y dy --Simplify before integrating--

π(9y - y²/2) |₀⁹ --Integrate--

π(9•9 - 9²/2) --Plug in upper bound (lower bound of y=0 gives value of 0)--

81π/2 --Simplify final answer--

Note that both methods result in the same final area. It's just based on what you are most comfortable working with.

Answer 2

Using the shell method, take a cross section of the region of width Δx that is parallel to the y-axis, has height x² and is at a distance of x from the y-axis.

Rotating the cross section about the y-axis generates a cylindrical shell with radius x, thickness Δx, and height x².

Volume of typical shell = 2πx(x²)Δx = 2πx³Δx

Volume of solid = V = 2π∫_{(0 to 3)} x³dx = (π/2)x⁴_{(0 to 3)} = 81π / 2