To determine

To find:

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

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

∫0c2cos⁡x-cos⁡(x-c)dx=-∫π2+cπcos⁡(x-c)dx

By solving integrals,

sin⁡x-sin⁡x-c0c2=-sin⁡x-cπ2+cπ

By substituting limits,

sin⁡c2-sin⁡c2-c-(-sin⁡-c)=-sin⁡π-c+sin⁡π2+c-c

By simplifying,

2sin⁡c2-sin⁡c=-sin⁡c+1

By simplifying, sin⁡c2=1/2 So

c=π3

Conclusion:

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