Respuesta :

wizard123
First find any intersection points, set 2 functions equal.
5cos(2x) = 5sin(4x) = 10sin(2x)cos(2x)

5cos(2x)(2sin(2x)-1) = 0

x = pi/4, pi/12

For interval 0 5sin(4x)

For interval pi/12 < x< pi/4, 5sin(4x) > 5 cos(2x)

Set up 2 integrals to express area for each interval.

[tex]A = 5\int_0^{\pi/12} cos(2x) - sin(4x) dx + 5 \int_{\pi/12}^{\pi/4} sin(4x) - cos(2x) dx[/tex]

Integrate using u-substitution

[tex]A = 5|_0^{\pi/12} [\frac{1}{2}sin(2x) +\frac{1}{4} cos(4x) ] + 5 |_{\pi/12}^{\pi/4} [-\frac{1}{4}cos(4x) - \frac{1}{2}sin(2x) ][/tex]

Evaluate limits

[tex]A = 5[(\frac{1}{4} +\frac{1}{8})-(\frac{1}{4}) ] + 5 [(\frac{1}{4} - \frac{1}{2})-(-\frac{1}{8}-\frac{1}{4}) ] \\ \\ A = \frac{5}{8} +\frac{5}{8} = \frac{5}{4} [/tex]

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