Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].
-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the horizontal line y = 2.
A) 1.87 $\pi$
B) 2.87 $\pi$
C) 3.87 $\pi$
D) 0.87 $\pi$

Question

Quizplus · Accepted Answer

The correct answer is A.The volume of the solid can be found by integrating the area of the cross-sections of the solid. The cross-sections are disks with radius $r = 1 - x^2$ and height $h = 2 - (1 - x^2) = 1 + x^2$. Therefore, the volume of the solid is$$V = \pi \int_0^1 (1 - x^2)^2 dx = \pi \int_0^1 (1 - 2x^2 + x^4) dx = \pi \left[ x - \frac{2x^3}{3} + \frac{x^5}{5} \right]_0^1 = \pi \left( 1 - \frac{2}{3} + \frac{1}{5} \right) = \pi \left( \frac{8}{15} \right) = \frac{8\pi}{15} \approx 1.87 \pi.$$

Consider the Region Between the Graph Of f(x)=1−x2f ( x ) = 1 - x ^ { 2 }f(x)=1−x2

Consider the Region Between the Graph Of $f ( x ) = 1 - x ^ { 2 }$