Chapter 5, Problem 10RE

a.

To determine

To find the total volume using Riemann sum approximation.

Answer to Problem 10RE

Total volume $=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^n\pi {\sin }^2{m}_1(\Delta x)}$ .

Explanation of Solution

Given:

The radius is given by $r=\sin \left(m_i\right)$ .

Formula used:

Surface of sphere were generated by revolving the graph of the function $f(x)=\sin x$ about the x-axis. The partition the interval into n subinterval of equal length $\Delta x$ . Then since the sphere with planes perpendicular to the x-axis at the partition points, cutting it like a round loaf of bread into n parallel sides of width $\Delta x$ . When n is large, each slice can be approximately by the cylinder of volume $\pi r^{2}h$ . $h$ is $\Delta x$ while r varies according to where are on the x-axis. A logical radius to choose for each cylinder is $f(m_1)=\sin (m_1)$ , where $m_1$ is the midpoint of the interval where the $i^{th}$ , slice interest the x-axis.

Calculation:

Given radius is $r=\sin \left(m_i\right)$ .

The volume of the sphere by using MRAM to sum the cylinder volumes,

Volume of one cylinder

  $\begin{array}{l}V=\pi r^{2}h\\ $ V=\pi {(\sin (m_i))}^2\Delta x$V=\pi {\sin }^2(m_i)\Delta x\end{array}$

Using Riemann sum approximation,

Total volume $=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^n\pi {\sin }^2{m}_1(\Delta x)}$ .

b.

To determine

To find function using numerical integration capability.

Answer to Problem 10RE

The function is ${\displaystyle {\int }_a^{b}f(x)dx}=NINT(f(x),x,a,b)$

Explanation of Solution

Given:

The radius is given by $r=\sin \left(m_i\right)$ .

Formula used:

The right-hand side of the equation is an approximation of the left-hand side.

Calculation:

Calculator has a numerical integration capability, which will denote as NINT. In particular, will use $NINT(f(x),x,a,b)$ to denote a calculator approximation of ${\displaystyle {\int }_a^{b}f(x)dx}$ .

  ${\displaystyle {\int }_a^{b}f(x)dx}=NINT(f(x),x,a,b)$ .

The understanding that the right-hand side of the equation is an approximation of the left-hand side.

c.

To determine

To find the value of function using the NINT.

The value of the function is $4.9348$ .

Answer to Problem 10RE

The value of the function is $4.9348$ .

Explanation of Solution

Given:

The given function $f(x)=\pi {\sin }^{2}x$ .

Formula used:

Calculator has a numerical integration capability, which will denote as NINT. In particular, will use $NINT(f(x),x,a,b)$ to denote a calculator approximation of ${\displaystyle {\int }_a^{b}f(x)dx}$ .

  ${\displaystyle {\int }_a^{b}f(x)dx}=NINT(f(x),x,a,b)$ .

Calculation:

The given is $f(x)=\pi {\sin }^{2}x$ .

  $\begin{array}{l}{\displaystyle {\int }_a^b\pi {\sin }^{2}xdx}=NINT(\pi {\sin }^{2}x,x,a,b)\\ {\displaystyle {\int }_a^b\pi {\sin }^{2}xdx}=\mathrm{4.9348.}\end{array}$