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Net displacement via the Net Change Theorem

A piston moves along a straight track with velocity v(t)=10sin⁡(t)+2v(t) = 10\sin(t) + 2 (in meters per second), where tt is measured in seconds.

Find the net displacement of the piston over the time interval [0,2π][0, 2\pi].

Hint

Use the Net Change Theorem: net displacement equals ∫02πv(t) dt\int_0^{2\pi} v(t)\, dt. Find an antiderivative of v(t)v(t), then evaluate it at the two endpoints and subtract.

Step-by-step solution

By the Net Change Theorem, the net displacement is ∫02π(10sin⁡(t)+2) dt\int_0^{2\pi} (10\sin(t) + 2)\, dt. An antiderivative of v(t)v(t) is s(t)=−10cos⁡(t)+2ts(t) = -10\cos(t) + 2t.

Evaluate: s(2π)=−10cos⁡(2π)+2(2π)=−10+4πs(2\pi) = -10\cos(2\pi) + 2(2\pi) = -10 + 4\pi. Also s(0)=−10cos⁡(0)+0=−10s(0) = -10\cos(0) + 0 = -10. So the net displacement is s(2π)−s(0)=(−10+4π)−(−10)=4πs(2\pi) - s(0) = (-10 + 4\pi) - (-10) = 4\pi meters.

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