Area between Curves: Two Cosines

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Find the area between . Equations of the cosine curves bounding the area

Equations of the cosine curves bounding the area

Draw a graph of

on paper without using your calculator. Then check your diagram by clicking “Next”.

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We need to add

to this diagram. Will its period be half or twice that of

It’s period will be half as long, or we can say that it oscillates twice as fast.

How will the amplitudes of

and

compare?

Since the coefficient in front of the cosine is one for each, they will have the same amplitude.

Add

to your diagram. Then check your diagram by clicking “Next”.

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Now shade the area to be integrated. Then check your work by clicking “Next”.

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How do we determine the area?

We integrate over these two regions using tiny rectangles as shown in green in this diagram.
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How wide are the rectangles?

Their width is dx

How do we determine their height?

We subtract the y-coordinate of the bottom from the y-coordinate of the top.

Set up the indefinite integral.