Chapter 14.5, Problem 28AYU

To determine

To find: The following for an integral $\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{{\displaystyle \int }}}\left(\text{cos}x\right)dx$.

a. What area does the integral represent?

Answer to Problem 28AYU

a. Area under the graph of $ƒ\left(x\right)=\text{ cos}x$ from $\frac{-\pi }{4}$ to $\frac{\pi }{4}$.

Explanation of Solution

Given:

$\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{{\displaystyle \int }}}\left(\text{cos}x\right)dx$

Calculation:

a. Area under the graph of $ƒ\left(x\right)=\text{ cos}x$ from $\frac{-\pi }{4}$ to $\frac{\pi }{4}$.

To determine

To find: The following for an integral $\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{{\displaystyle \int }}}\left(\text{cos}x\right)dx$.

b. Provide a graph that illustrates this area.

Answer to Problem 28AYU

b.

Explanation of Solution

Given:

$\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{{\displaystyle \int }}}\left(\text{cos}x\right)dx$

Calculation:

b. Graph:

To determine

To find: The following for an integral $\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{{\displaystyle \int }}}\left(\text{cos}x\right)dx$.

c. Use a graphing utility to approximate this area.

Answer to Problem 28AYU

c. The area of the integral is $1.414$.

Explanation of Solution

Given:

$\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{{\displaystyle \int }}}\left(\text{cos}x\right)dx$

Calculation:

c. Use a graphing utility to approximate the area of the integral.

That is evaluate the integral,

The value of the integral $\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{{\displaystyle \int }}}\left(\text{cos}x\right)dx$ is $1.414$.

So, the area under the graph of $f$ from $\frac{-\pi }{4}$ to $\frac{\pi }{4}$ is $1.414$.