Chapter 6.2, Problem 42E

To determine

Find the volume of the described solid S.

Answer to Problem 42E

The volume of the described solid S is $2{\text{ units}}^3$

Explanation of Solution

Given information: Given curves $y=1-x^2$

Calculation:

  

The cross section (red line segment) represents the side of the square.

To solve the equation for $x$

  $\begin{array}{l}y=1-x^2\\ x^2=1-y\\ x=\pm \sqrt{1-y}\end{array}$

We will use the ‘ +’ one, which represents the right half of the parabola.

So the length that we get form it will have to be doubled. The area of one square

Will be

  $A={\left(2\sqrt{1-y}\right)}^2=4\left(1-y\right)$

Integrate from $0\text{ to }1$

  $\begin{array}{l}V={\displaystyle \underset{0}{\overset{1}{\int }}4\left(1-y\right)}dy\\ ={\left[y-\frac{1}{2}{y}^2\right]}_0^1\\ =\left[1-\frac{1}{2}-\left(0-0\right)\right]\\ =4\frac{1}{2}\\ =2{\text{ units}}^3\end{array}$

Thus the volume of the described solid S is $2{\text{ units}}^3$