Q. 42 - Calculus

Question

Consider the region between $f\left(x\right)=4-x^2$ and the line y = 5 on [0, 2]. For each line of rotation given in Exercises 39–42, use the shell method to construct definite integrals to find the volume of the resulting solid.

Step-by-Step Solution

Answer

The integral can be given as $V=2\pi {\int }_2^0\left(\right(5-y\left)\right(2-\sqrt{4-y})dy$ and value of integral is $4\pi$.

Step 1: Given information

We are given $f\left(x\right)=4-x^2$ and

Step 2: Find the integral and evaluate it

We know that integral can be given as

$V=2\pi {\int }_c^{d}r\left(y\right)h\left(y\right)dy$  where r and h are the radius and height respectively

the axis of revolution is y=5 Hence the radius can be given as $r\left(y\right)=5-y$ and the height can be given as $h\left(y\right)=(2-\sqrt{4-y})$

substituting the values in integral we get,

$$\begin{gathered} V=2\pi {\int }_2^0\left(\right(5-y\left)\right(2-\sqrt{4-y})dy \\ V=2\pi {\int }_2^0(10-5\sqrt{4-y}-2y+y\sqrt{4-y})dy \\ V=4\pi \end{gathered}$$