Chapter 14.5, Problem 27AYU

To determine

To find: The following for an integral $\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\left(\sin x\right)dx$.

a. What area does the integral represent?

Answer to Problem 27AYU

a. Area under the graph of $ƒ\left(x\right)=\text{ sin}x$ from 0 to $\frac{\pi }{2}$.

Explanation of Solution

Given:

$\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\left(\sin x\right)dx$

Calculation:

a. Area under the graph of $ƒ\left(x\right)=\text{ sin}x$ from 0 to $\frac{\pi }{2}$.

To determine

To find: The following for an integral $\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\left(\sin x\right)dx$.

b. Provide a graph that illustrates this area.

Answer to Problem 27AYU

b.

Explanation of Solution

Given:

$\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\left(\sin x\right)dx$

Calculation:

b. Graph:

To determine

To find: The following for an integral $\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\left(\sin x\right)dx$.

c. Use a graphing utility to approximate this area.

Answer to Problem 27AYU

c. The area of the integral is 1.

Explanation of Solution

Given:

$\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\left(\sin x\right)dx$

Calculation:

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

That is evaluate the integral,

The value of the integral $\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\left(\sin x\right)dx$ is 1.

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