Find the area of the region(s) between the given curves below on the given interval.

Question

y = 5 cos x,  y = 5 − 5 cos x

 from x = 0 to x = 𝜋

Answer 1

Draw a sketch.

The curves intersect when 5cosx = 5 - 5cosx. So, cosx = 1/2.

Between x = 0 and x = π, cosx = 1/2 at x = π/3

Between x = 0 and x = π/3, the upper boundary is y = 5cosx and y = 5-5cosx is the lower boundary.

Between x = π/3 and π, the boundaries are reversed.

Area = ∫_{(0 to pi/3)} (5cosx - 5 + 5cosx)dx + ∫_{(pi/3 to pi)} (5 - 5cosx - 5cosx)dx

= ∫_{(0 to pi/3)} (10cosx - 5)dx + ∫_{(pi/3 to pi)} (5 - 10cosx)dx

^Evaluate.

^

Answer 2

We are given two curves, we must find where they intersect to find the bounds of integration. Setting them equal to each other, we find that at x=arccos(1/2) is a point where the curves intersect (and the one previously on the top changes to be on the bottom). Thus, the region is broken into two integrals, the first from x=0 to x=arccos(1/2) and the second from x=arccos(1/2) to x=pi. Then, all we must do is subtract the top function from the bottom in both integrals (the order will be the opposite as mentioned before): this is our integrand. Integrate both integrands on their respective bounds and the total area is their sum.