Question

sketch the region enclosed by the given curves and find its area: y=cosx, y=2-cosx, -pi/3<=x<=pi/3

Answer 1

thats easy

Answer 2

2-cos(x) is above cos(x) over the domain

Answer 3

\[\int\limits_{-\frac{\pi}{3} }^{\frac{\pi}{3} } ( 2-\cos(x) - \cos(x))dx \]

Answer 4

So to do this, you need to find the area between the curves. The easiest way is by using integration. You are also given the domain of the function, so those are the bounds of your integral.
\[\int\limits_{?}^{?}y _{1}-y _{2}dx\]
Here your bounds are -pi/3 and pi/3
Your \[y_{2}=cos(x)\]
and \[y_{1}=2-cos(x)\]
You are subtracting the lower bound from the upper bound, and since 2-cos(x) is always above cos(x) it becomes 2-2cos(x).
So your final integral is
\[\int\limits_{-pi/3}^{pi/3}(2-cos(x))-(cos(x))dx\]
which is \[\int\limits_{-\pi/3}^{\pi/3}2-2\cos(x)dx\]
Which comes out to
.725

Answer 5

or
\[-2\sqrt{3}+4\pi/3\]