#### To determine

**To find:**

The value of c such that the area of the region enclosed by the curves y=cosx, y=cosx-c, and x=0 is equal to the area of the region enclosed by the curves y=cosx-c, x=π, and y=0.

#### Answer

π3

#### Explanation

**1) Concept:**

Put the two functions equal to each other, and find the point of intersection. These points are the integral limits.

**2) Given:**

0<c<π2

3) **Calculation:**

At the point of intersection of the curves,cosx=cosx-c. Since x is in [0,π]. It follows that x=-(x-c) or x=-(x-c). But c is non-zero. So, x=c2

The point where cos(x-c) crosses the x- axis, cos(x-c)=0 so x-c=π2. Hence x=π2+c

The value of c such that the area of the region enclosed by the curves y=cosx, y=cosx-c, and x=0 is equal to the area of the region enclosed by the curves y=cosx-c, x=π, and y=0:

Equate the expression for both areas. Since 0>cos(x-c) in [π2+c,π] and cosx>cos(x-c) in [0,c2] we have

∫0c2cosx-cos(x-c)dx=-∫π2+cπcos(x-c)dx

By solving integrals,

sinx-sinx-c0c2=-sinx-cπ2+cπ

By substituting limits,

sinc2-sinc2-c-(-sin-c)=-sinπ-c+sinπ2+c-c

By simplifying,

2sinc2-sinc=-sinc+1

By simplifying, sinc2=1/2 So

c=π3

**Conclusion:**

Therefore, if the value of c=π3 then the area of the region enclosed by the curves y=cosx, y=cosx-c, and x=0 is equal to the area of the region enclosed by the curves y=cosx-c, x=π, and y=0.