Chapter 8.3, Problem 1QQ

To determine

To Find: the volumeof a solid is the region in the first quadrant bounded by the $x-axis$ , the graph of $y={\sin }^{-1}x$ and the vertical line $x=1$

Answer to Problem 1QQ

  $(C)$ $\text{Volume}\approx \text{0}\text{.467}$

Explanation of Solution

Given information: The base of a solid is the region in the first quadrant bounded by the $x-axis$ , the graph of $y={\sin }^{-1}x$ and the vertical line $x=1$

The volume of a solid of known integrable cross-section area $A(x)$ from $x=a$ to $x=b$ is the integral of Afrom a to b

  $V={\displaystyle \underset{a}{\overset{b}{\int }}A(x)dx}$

Here,

  $A(x)={\left({\sin }^{-1}x\right)}^2$ and $a=0,b=1$

  $\begin{array}{l}V={\displaystyle \underset{0}{\overset{1}{\int }}{\left({\sin }^{-1}x\right)}^{2}dx}\\ V={\left[{\left({\sin }^{-1}x\right)}^{2}x+2\left({\sin }^{-1}x\right)\sqrt{1-x^2}-2x\right]}_0^1\\ V=\frac{{\pi }^2}{4}-2\\ V\approx 0.467\end{array}$