Chapter 7.2, Problem 59E

To determine

To show: It needs to be determined that the given theorem is true.

Answer to Problem 59E

The given theorem is proved to be true as the entire summation remains same.

Explanation of Solution

Given information: A Cavalieri’s Theorem is given which states that 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 as shown in figure in the textbook and then the regions have the same area.

Calculation: The entire region can be divided into infinite number of cross sections with a width of $\Delta x$ and a vertical length of $f\left(x_i\right)$, then the total area of the region will be given by:

  $A=\underset{n\to \infty }{\lim }\Delta x{\displaystyle \sum _{i=1}^{n}f\left(x_i\right)}$

Hence when the vertical cross section remains same then the $f\left(x_i\right)$ also remains same which means that entire summation remains same.

Thus, the area is same.