Calculus with Vector Functions — Question 10

Question 10

Evaluate $$\int_0^\pi\left\langle t\sin t,\,t\cos t,\,\sin^2t\right\rangle\,dt.$$ Then interpret the result as displacement if the integrand is velocity, and find the average velocity on $[0,\pi]$.

Question 10 – Solution

Strategy Integrate componentwise, using integration by parts for the first two components.

Integral We have $$\int_0^\pi t\sin t\,dt=\pi,\qquad \int_0^\pi t\cos t\,dt=-2,\qquad \int_0^\pi\sin^2t\,dt=\frac\pi 2.$$ Therefore the displacement is $\boxed{\left\langle\pi,-2,\pi/2\right\rangle}$.

Average velocity Divide displacement by the elapsed time $\pi$: $$\boxed{\vec v_{\mathrm{avg}}=\left\langle 1,-2/\pi,1/2\right\rangle}.$$

Verification Multiplying the average velocity by $\pi$ recovers the net displacement.