Consider the region between the graph of $f ( x ) = 1 - x ^ { 2 }$ and the line y = 2 on the interval [0, 1].
-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.
A) $\pi$
B) 2 $\pi$
C) 1.5 $\pi$
D) 2.5 $\pi$

Question

Quizplus · Accepted Answer

The volume of the solid is given by the integral $\pi \int_0^1 (2 - (1 - x^2))^2 dx$. Evaluating this integral gives $\pi \int_0^1 (1 + 2x^2 - x^4) dx = \pi \left[ x + \frac{2x^3}{3} - \frac{x^5}{5} \right]_0^1 = \frac{3\pi}{5}$.

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 $f ( x ) = 1 - x ^ { 2 }$