Find the volume of the solid bounded above by the surface and below by the plane region R. and R is the region bounded by the graphs of .

B)
C)
D)
E)

Question

Find the volume of the solid bounded above by the surface  and below by the plane region R. ​  and R is the region bounded by the graphs of  .
​
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

Using the washer method, the volume of the solid can be found by:
$$V=\int_{0}^{1} \pi\left[(2-x^2)^2-(x^2-1)^2ight]dx$$
Simplifying the integral and solving:
$$\begin{aligned} V&=\int_{0}^{1} \pi \left[4-5x^4+x^8ight]dx\ &=\pi\left[4x-\frac{5}{5}x^5+\frac{1}{9}x^9ight]_{0}^{1}\ &=\pi\left(4-\frac{5}{5}+\frac{1}{9}ight)\ &=\frac{28\pi}{9} \end{aligned}$$
Therefore, the answer is A.

Find the Volume of the Solid Bounded Above by the Surface