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blsea [12.9K]
3 years ago
7

Consider the area under one arch of the curve y(t) = ksin(wt) for t ≥ 0 where k and w are positive constants.

Mathematics
1 answer:
Stolb23 [73]3 years ago
5 0
The period of the function is \frac{2\pi}{w} so the first arch extends over the interval \left[ 0,\frac{\pi}{w} \right]. The area is

\int_0^{\pi/w} \sin{(wt)} dt

The integral can be found by substitution.
u=wt \\ du=w \; dt \\ \frac{du}{u}=dt

The limits of integration change, too.

t=0 \Rightarrow u=0 \\ t=\frac{\pi}{w} \Rightarrow u=\pi

\frac{k}{w} \int_0^\pi \sin{u} \; du = \frac{k}{w} \left[-\cos{u} \right]_0^\pi = \frac{k}{w} \left[ 1-(-1) \right] = \frac{2k}{w}

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