   Chapter 4.5, Problem 54E

Chapter
Section
Textbook Problem

# Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area. y = 2 sin x − sin 2 x ,     0 ≤ x ≤ π

To determine

To estimate:

Area of the region that lies under the given curve using graph and then find the exact area

Explanation

1) Concept:

To find exact area, integrate given function.

2) formula:

sinx=-cosx

3) Given:

y=2sinx-sin2x , 0xπ

4) Graph:

5) Calculation:

From the graph, theshaded region is approximately a triangle with the base π and the height 2.6. So, its area is approximately 12 · π ·2.6=4.084

Now to find exact area, integrate given function.

A=0π(2sinx-sin2x)dx

Simplify.

=0π2sinxdx-  0πsin2x dx

=20πsinxdx-  0πsin2x dx

=2-cosx0π-  0πsin2x dx

For second integral, substitute 2x=u

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