#### To determine

**To estimate:**

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

#### Answer

4

#### Explanation

**1) Concept:**

To find exact area, integrate given function.

**2) formula:**

∫sinx=-cosx

**3) Given:**

y=2sinx-sin2x , 0≤x≤π

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

=2∫0πsinxdx- ∫0πsin2x dx

=2-cosx0π- ∫0πsin2x dx

For second integral, substitute 2x=u.

Differentiating with respect to x

2 dx=du

dx=12 du

The limits change and the new limits of integration are calculated by substituting

for x=0, u=2(0)=0 and for x=π, u=2(π) =2π

Therefore, the given integral becomes

=2-cosx0π- ∫02π12sinu du

=2-cosx0π- 12∫02πsinu du

=2-cosx0π- 12-cosu02π

=-2cosx0π+ 12cosu02π

=-2cosπ-(cos0)+12cos2π-(cos0)

=-2-1-1+121-1

=-2-2+120

Simplify.

=4

Therefore, thearea of the region that lies under the given curveis 4.

**Conclusion:**

Therefore, the area of the region that lies under the given curveis 4.