Chapter 8.5, Problem 13E

a.

To determine

Explain to approximate the force against the end of the tank by a Reimann sum.

Answer to Problem 13E

  ${\displaystyle \sum _{i=1}^{k}62.4\cdot 2x_i\cdot \sqrt{9-x_i^2}}dx$ approximately the force against the end of the tank by a Reimann sum.

Explanation of Solution

Given information: The vertical end of a tank containing water weighing $62.4lb/f{t}^3$ .

Shape- semicircle.

  

Calculation:

The force of the water against the end of the tank can be approximated by diving the height into k partitions and evaluating the Riemann sum,

  ${\displaystyle \sum _{i=1}^{k}62.4\cdot 2x_i\cdot \sqrt{9-x_i^2}}dx$

Hence, ${\displaystyle \sum _{i=1}^{k}62.4\cdot 2x_i\cdot \sqrt{9-x_i^2}}dx$ approximately the force against the end of the tank by a Reimann sum.

b.

To determine

To find the force as an integral and evaluate it.

Answer to Problem 13E

The required solution is 1123.2.

Explanation of Solution

Given information: The vertical end of a tank containing water weighing $62.4lb/f{t}^3$ .

Shape- semicircle.

  

Calculation:

As per the given problem,

Which by taking the limit as $k\to \infty$ , gives us the integral,

  $\begin{array}{l}{\displaystyle {\int }_0^{3}62.4\cdot 2\cdot x\sqrt{9-x^2}dx}\\ =1123.2\end{array}$

Hence, the required solution is 1123.2.