Math Problem Solver

Find the area of the region enclosed by the curves yequals2 cosine x and yequals2 cosine font size decreased by 8 2 x for 0less than or equalsxless than or equalspi.

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y equals 2 cosine xy equals 2 cosine font size decreased by 8 2 x

A coordinate system has a horizontal x-axis from less than 0 to pi plus in increments of StartFraction pi over 4 EndFraction and a vertical y-axis labeled from less than negative 2 to 2 plus in increments of 2. From left to right, a curve labeled y equals 2 cosine x rises at a decreasing rate to a maximum at left parenthesis 0 comma 2 right parenthesis comma falls at an increasing rate to the point left parenthesis StartFraction pi over 2 EndFraction comma 0 right parenthesis comma falls at a decreasing rate to a minimum at left parenthesis pi comma negative 2 right parenthesis comma and rises at an increasing rate. A second curve labeled y equals 2 cosine 2 x repeats with a period of pi. Within one period from left to right, the curve starts at a maximum at left parenthesis 0 comma 2 right parenthesis comma falls at an increasing rate to left parenthesis StartFraction pi over 4 EndFraction comma 0 right parenthesis comma falls at a decreasing rate to a minimum at left parenthesis StartFraction pi over 2 EndFraction comma negative 2 right parenthesis comma rises at an increasing rate to left parenthesis StartFraction 3 pi over 4 EndFraction comma 0 right parenthesis comma and rises at a decreasing rate to a maximum at left parenthesis pi comma 2 right parenthesis. The regions bounded by the curves, to the right of the y-axis and to the left of x = pi are shaded.

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Part 1

The area of the region enclosed by the curves is

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​(Type an exact​ answer, using radicals as​ needed.)