Chapter 14.5, Problem 22AYU

To determine

To solve: The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$,

a. Graph $f$, indicating the area $A$ under $f$ from 0 to $\frac{\pi }{2}$.

Answer to Problem 22AYU

a.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$.

Calculation:

$ƒ\left(0\right)=\text{cos}{0}^0=1$; $ƒ\left(\frac{\pi }{16}\right)=\text{cos}{11.25}^0=0.981$; $ƒ\left(\frac{\pi }{8}\right)=\text{ cos}{22.5}^0=0.924$; $ƒ\left(\frac{3\pi }{16}\right)=\text{ cos33}{\text{.75}}^0=\text{ 0}\text{.831}$; $ƒ\left(\frac{\pi }{4}\right)=\text{ cos4}{5}^0=0.707$; $ƒ\left(\frac{5\pi }{16}\right)=\text{ cos}{56.25}^0=0.556$; $ƒ\left(\frac{3\pi }{8}\right)=\text{ cos67}{.5}^0=0.383$; $ƒ\left(\frac{7\pi }{16}\right)=\text{ cos78}{.75}^0=0.195$

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

To determine

To solve: The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$,

b. Approximate the area $A$ by Partition $\left[0,\frac{\pi }{2}\right]$ into four subintervals of equal length and choose u as the left endpoint of each subinterval.

Answer to Problem 22AYU

b. The area $A$ is approximated as $2.785$.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$.

Calculation:

$ƒ\left(0\right)=\text{cos}{0}^0=1$; $ƒ\left(\frac{\pi }{16}\right)=\text{cos}{11.25}^0=0.981$; $ƒ\left(\frac{\pi }{8}\right)=\text{ cos}{22.5}^0=0.924$; $ƒ\left(\frac{3\pi }{16}\right)=\text{ cos33}{\text{.75}}^0=\text{ 0}\text{.831}$; $ƒ\left(\frac{\pi }{4}\right)=\text{ cos4}{5}^0=0.707$; $ƒ\left(\frac{5\pi }{16}\right)=\text{ cos}{56.25}^0=0.556$; $ƒ\left(\frac{3\pi }{8}\right)=\text{ cos67}{.5}^0=0.383$; $ƒ\left(\frac{7\pi }{16}\right)=\text{ cos78}{.75}^0=0.195$

b. Partition $\left[0,\frac{\pi }{2}\right]$ into four subintervals of equal length $\frac{\pi }{8}$ and choose u as the left endpoint of each subinterval.

The area $A$ is approximated as

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

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

$=2.785$

To determine

To solve: The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$,

c. Approximate the area $A$ by Partition $\left[0,\frac{\pi }{2}\right]$ into eight subintervals of equal length and choose u as the left endpoint of each subinterval.

Answer to Problem 22AYU

c. The area $A$ is approximated as $5.471$.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$.

Calculation:

$ƒ\left(0\right)=\text{cos}{0}^0=1$; $ƒ\left(\frac{\pi }{16}\right)=\text{cos}{11.25}^0=0.981$; $ƒ\left(\frac{\pi }{8}\right)=\text{ cos}{22.5}^0=0.924$; $ƒ\left(\frac{3\pi }{16}\right)=\text{ cos33}{\text{.75}}^0=\text{ 0}\text{.831}$; $ƒ\left(\frac{\pi }{4}\right)=\text{ cos4}{5}^0=0.707$; $ƒ\left(\frac{5\pi }{16}\right)=\text{ cos}{56.25}^0=0.556$; $ƒ\left(\frac{3\pi }{8}\right)=\text{ cos67}{.5}^0=0.383$; $ƒ\left(\frac{7\pi }{16}\right)=\text{ cos78}{.75}^0=0.195$

c. Partition $\left[0,\frac{\pi }{2}\right]$ into eight subintervals of equal length $\frac{\pi }{16}$ and choose u as the left endpoint of each subinterval.

The area $A$ is approximated as

$A=f\left(0\right)\frac{\pi }{16}+f\left(\frac{\pi }{16}\right)\frac{\pi }{16}+f\left(\frac{\pi }{8}\right)\frac{\pi }{16}+f\left(\frac{3\pi }{16}\right)\frac{\pi }{16}+f\left(\frac{\pi }{4}\right)\frac{\pi }{16}+f\left(\frac{5\pi }{16}\right)\frac{\pi }{16}+f\left(\frac{3\pi }{8}\right)\frac{\pi }{16}+f\left(\frac{7\pi }{16}\right)\frac{\pi }{16}$

$=1\left(0.981\right)+\left(0.981\right)\left(0.981\right)+0.924\left(0.981\right)+\left(0.831\right)\left(0.981\right)+\left(0.707\right)\left(0.981\right)+\left(0.556\right)\left(0.981\right)+\left(0.383\right)\left(0.195\right)+\left(0.195\right)\left(0.981\right)$

$=5.471$

To determine

To solve: The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$,

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

Answer to Problem 22AYU

d. The area $A$ as an integral is $\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\text{cos}xdx$.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$.

Calculation:

$ƒ\left(0\right)=\text{cos}{0}^0=1$; $ƒ\left(\frac{\pi }{16}\right)=\text{cos}{11.25}^0=0.981$; $ƒ\left(\frac{\pi }{8}\right)=\text{ cos}{22.5}^0=0.924$; $ƒ\left(\frac{3\pi }{16}\right)=\text{ cos33}{\text{.75}}^0=\text{ 0}\text{.831}$; $ƒ\left(\frac{\pi }{4}\right)=\text{ cos4}{5}^0=0.707$; $ƒ\left(\frac{5\pi }{16}\right)=\text{ cos}{56.25}^0=0.556$; $ƒ\left(\frac{3\pi }{8}\right)=\text{ cos67}{.5}^0=0.383$; $ƒ\left(\frac{7\pi }{16}\right)=\text{ cos78}{.75}^0=0.195$

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

The area $A$ as an integral is $\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\text{cos}xdx$.

To determine

To solve: The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$,

e. Use a graphing utility to approximate the integral.

Answer to Problem 22AYU

e. The area under the graph of $f$ from 0 to $\frac{\pi }{2}$ is 1.

Explanation of Solution

Given:

The function $ƒ\left(x\right)=\text{cos}x$ is defined on the interval $\left[0,\frac{\pi }{2}\right]$.

Calculation:

$ƒ\left(0\right)=\text{cos}{0}^0=1$; $ƒ\left(\frac{\pi }{16}\right)=\text{cos}{11.25}^0=0.981$; $ƒ\left(\frac{\pi }{8}\right)=\text{ cos}{22.5}^0=0.924$; $ƒ\left(\frac{3\pi }{16}\right)=\text{ cos33}{\text{.75}}^0=\text{ 0}\text{.831}$; $ƒ\left(\frac{\pi }{4}\right)=\text{ cos4}{5}^0=0.707$; $ƒ\left(\frac{5\pi }{16}\right)=\text{ cos}{56.25}^0=0.556$; $ƒ\left(\frac{3\pi }{8}\right)=\text{ cos67}{.5}^0=0.383$; $ƒ\left(\frac{7\pi }{16}\right)=\text{ cos78}{.75}^0=0.195$

e. Use a graphing utility to approximate the integral.

That is evaluate the integral,

The value of the integral $\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\text{cos}xdx$ is 1.

So the area under the graph of $f$ from 0 to $\frac{\pi }{2}$ is 1.