Chapter 8.2, Problem 57E

To determine

To prove: If two plane regions can be arranged to lie over the same interval of the $x-axis$ in such a way that they have identical vertical cross sections at every point, then the regions have the same area

Explanation of Solution

Given information: Cavalieri’s Theorem

Formula used:If $f(x)$ and $g(x)$ are continuous with $f(x)\ge g(x)$ throughout $[a,b]$ , then the area between the curves $y=f(x)$ and $y=g(x)$ from a to b is the integral of $[f(x)-g(x)]$ from a to b

  $A={\displaystyle \underset{a}{\overset{b}{\int }}[f(x)-g(x)]}dx$

Proof:

Since $f(x)-g(x)$ is the same for each region where $f(x)$ and $g(x)$ represent the upper and lower edges,

  $A={\displaystyle \underset{a}{\overset{b}{\int }}[f(x)-g(x)]}dx$

Will be the same for each