Math Challenges

Submissions for Problem #14

Find the volume of the solid obtained by rotating the region bounded by these curves about the y-axis.

\[ \displaylines{y=x^{\frac23}\\ x=1\\ y=0} \]

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Solution: \( \displaylines{\int_0^1\pi\left(1\right)^2-\pi\left(y^{\frac32}\right)^2dy=\\ \pi-\pi\int_0^1y^3dy=\pi-\pi\left(\frac14\right)=\frac{3\pi}{4}} \)

Explanation:
We take an outer minus inner volume, using length x as the radius. The "washer method".