Chapter 14.5, Problem 21AYU

To determine

To solve: The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$,

a. Graph $f$, indicating the area $A$ under $f$ from 0 to $\pi$.

Answer to Problem 21AYU

a.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$.

Calculation:

$ƒ\left(0\right)=\text{sin}{0}^0=0$; $ƒ\left(\frac{\pi }{8}\right)=\text{sin}{22.5}^0=0.383$; $ƒ\left(\frac{\pi }{4}\right)=\text{sin}\text{4}{5}^0=0.707$; $ƒ\left(\frac{3\pi }{8}\right)=\text{sin}{67.5}^0=0.924$; $ƒ\left(\frac{\pi }{2}\right)=\text{sin}{\text{90}}^0=\text{ 1}$; $ƒ\left(\frac{5\pi }{8}\right)=\text{sin}\text{112}{.5}^0=0.924$; $ƒ\left(\frac{3\pi }{4}\right)=\text{sin}\text{13}{5}^0=0.707$; $ƒ\left(\frac{7\pi }{8}\right)=\text{sin}\text{157}{.5}^0=0.383$

a. Graph $ƒ\left(x\right)=\text{sin}x$

To determine

To solve: The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$,

b. Approximate the area $A$ by Partition $\left[0,\pi \right]$ into four subintervals of equal length and choose u as the left endpoint of each subinterval.

Answer to Problem 21AYU

b. The area $A$ is approximated as $1.706$.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$.

Calculation:

$ƒ\left(0\right)=\text{sin}{0}^0=0$; $ƒ\left(\frac{\pi }{8}\right)=\text{sin}{22.5}^0=0.383$; $ƒ\left(\frac{\pi }{4}\right)=\text{sin}\text{4}{5}^0=0.707$; $ƒ\left(\frac{3\pi }{8}\right)=\text{sin}{67.5}^0=0.924$; $ƒ\left(\frac{\pi }{2}\right)=\text{sin}{\text{90}}^0=\text{ 1}$; $ƒ\left(\frac{5\pi }{8}\right)=\text{sin}\text{112}{.5}^0=0.924$; $ƒ\left(\frac{3\pi }{4}\right)=\text{sin}\text{13}{5}^0=0.707$; $ƒ\left(\frac{7\pi }{8}\right)=\text{sin}\text{157}{.5}^0=0.383$

b. Partition $\left[0,\pi \right]$ into four subintervals of equal length $\frac{\pi }{4}$ and choose u as the left endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(0\right)\frac{\pi }{4}+f\left(\frac{\pi }{4}\right)\frac{\pi }{4}+f\left(\frac{\pi }{2}\right)\frac{\pi }{4}+f\left(\frac{3\pi }{4}\right)\frac{\pi }{4}$

$=0\left(0.707\right)+0.707\left(0.707\right)+1\left(0.707\right)+0.707\left(0.707\right)$

$=1.706$

To determine

To solve: The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$,

c. Approximate the area $A$ by Partition $\left[0,\pi \right]$ into eight subintervals of equal length and choose u as the left endpoint of each subinterval.

Answer to Problem 21AYU

c. The area $A$ is approximated as $1.779$.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$.

Calculation:

$ƒ\left(0\right)=\text{sin}{0}^0=0$; $ƒ\left(\frac{\pi }{8}\right)=\text{sin}{22.5}^0=0.383$; $ƒ\left(\frac{\pi }{4}\right)=\text{sin}\text{4}{5}^0=0.707$; $ƒ\left(\frac{3\pi }{8}\right)=\text{sin}{67.5}^0=0.924$; $ƒ\left(\frac{\pi }{2}\right)=\text{sin}{\text{90}}^0=\text{ 1}$; $ƒ\left(\frac{5\pi }{8}\right)=\text{sin}\text{112}{.5}^0=0.924$; $ƒ\left(\frac{3\pi }{4}\right)=\text{sin}\text{13}{5}^0=0.707$; $ƒ\left(\frac{7\pi }{8}\right)=\text{sin}\text{157}{.5}^0=0.383$

c. Partition $\left[0,\pi \right]$ into eight subintervals of equal length $\frac{\pi }{8}$ and choose u as the left endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(0\right)\frac{\pi }{8}+f\left(\frac{\pi }{8}\right)\frac{\pi }{8}+f\left(\frac{\pi }{4}\right)\frac{\pi }{8}+f\left(\frac{3\pi }{8}\right)\frac{\pi }{8}+f\left(\frac{\pi }{2}\right)\frac{\pi }{8}+f\left(\frac{5\pi }{8}\right)\frac{\pi }{8}+f\left(\frac{3\pi }{4}\right)\frac{\pi }{8}+f\left(\frac{7\pi }{8}\right)\frac{\pi }{2}$

$=0\left(0.383\right)+0.383\left(0.383\right)+0.707\left(0.383\right)+0.924\left(0.383\right)+1\left(0.383\right)+0.924\left(0.383\right)+0.707\left(0.383\right)$

$=1.779$

To determine

To solve: The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$,

d. Express the area $A$ as an integral.

Answer to Problem 21AYU

d. The area $A$ as an integral is $\underset{0}{\overset{\pi }{{\displaystyle \int }}}\text{sin}xdx$.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$.

Calculation:

$ƒ\left(0\right)=\text{sin}{0}^0=0$; $ƒ\left(\frac{\pi }{8}\right)=\text{sin}{22.5}^0=0.383$; $ƒ\left(\frac{\pi }{4}\right)=\text{sin}\text{4}{5}^0=0.707$; $ƒ\left(\frac{3\pi }{8}\right)=\text{sin}{67.5}^0=0.924$; $ƒ\left(\frac{\pi }{2}\right)=\text{sin}{\text{90}}^0=\text{ 1}$; $ƒ\left(\frac{5\pi }{8}\right)=\text{sin}\text{112}{.5}^0=0.924$; $ƒ\left(\frac{3\pi }{4}\right)=\text{sin}\text{13}{5}^0=0.707$; $ƒ\left(\frac{7\pi }{8}\right)=\text{sin}\text{157}{.5}^0=0.383$

d. Express the area $A$ as an integral.

The area $A$ as an integral is $\underset{0}{\overset{\pi }{{\displaystyle \int }}}\text{sin}xdx$.

To determine

To solve: The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$,

e. Use a graphing utility to approximate the integral.

Answer to Problem 21AYU

e. The area under the graph of $f$ from 0 to $\pi$ is $6.326$.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{sin}x$ is defined on the interval $\left[0,\pi \right]$.

Calculation:

$ƒ\left(0\right)=\text{sin}{0}^0=0$; $ƒ\left(\frac{\pi }{8}\right)=\text{sin}{22.5}^0=0.383$; $ƒ\left(\frac{\pi }{4}\right)=\text{sin}\text{4}{5}^0=0.707$; $ƒ\left(\frac{3\pi }{8}\right)=\text{sin}{67.5}^0=0.924$; $ƒ\left(\frac{\pi }{2}\right)=\text{sin}{\text{90}}^0=\text{ 1}$; $ƒ\left(\frac{5\pi }{8}\right)=\text{sin}\text{112}{.5}^0=0.924$; $ƒ\left(\frac{3\pi }{4}\right)=\text{sin}\text{13}{5}^0=0.707$; $ƒ\left(\frac{7\pi }{8}\right)=\text{sin}\text{157}{.5}^0=0.383$

e. Use a graphing utility to approximate the integral.

That is evaluate the integral,

The value of the integral $\underset{0}{\overset{\pi }{{\displaystyle \int }}}\text{sin}xdx$ is $6.326$.

So the area under the graph of $f$ from 0 to $\pi$ is $6.326$.