91Ó°ÊÓ

Displacement from a table of velocities The velocities (in \(\mathrm{mi} / \mathrm{hr}\) ) of an automobile moving along a straight highway over a two- hour period are given in the following table.$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline t \text { (hr) } & 0 & 0.25 & 0.5 & 0.75 & 1 & 1.25 & 1.5 & 1.75 & 2 \\\ \hline v \text { (mi/hr) } & 50 & 50 & 60 & 60 & 55 & 65 & 50 & 60 & 70 \\ \hline\end{array}$$,a. Sketch a smooth curve passing through the data points. b. Find the midpoint Riemann sum approximation to the displacement on [0,2] with \(n=2\) and \(n=4\).

Find or approximate all points at which the given function equals its average value on the given interval. \(f(x)=\frac{\pi}{4} \sin x\) on \([0, \pi]\)

Net area from graphs The accompanying figure shows four regions bounded by the graph of \(y=x \sin x: R_{1}, R_{2}, R_{3},\) and \(R_{4},\) whose areas are \(1, \pi-1, \pi+1,\) and \(2 \pi-1,\) respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. $$\int_{0}^{3 \pi / 2} x \sin x d x$$

Displacement from a table of velocities The velocities (in \(\mathrm{m} / \mathrm{s}\) ) of an automobile moving along a straight freeway over a four second period are given in the following table.$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline t(\mathrm{s}) & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\ \hline v(\mathrm{m} / \mathrm{s}) & 20 & 25 & 30 & 35 & 30 & 30 & 35 & 40 & 40 \\\ \hline\end{array}$$,a. Sketch a smooth curve passing through the data points. b. Find the midpoint Riemann sum approximation to the displacement on [0,4] with \(n=2\) and \(n=4\) subintervals.

Find the following integrals. $$\int(z+1) \sqrt{3 z+2} d z$$

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{4} x(x-2)(x-4) d x$$

Find or approximate all points at which the given function equals its average value on the given interval. $$f(x)=1-|x| \text { on }[-1,1]$$

Net area from graphs The accompanying figure shows four regions bounded by the graph of \(y=x \sin x: R_{1}, R_{2}, R_{3},\) and \(R_{4},\) whose areas are \(1, \pi-1, \pi+1,\) and \(2 \pi-1,\) respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. $$\int_{0}^{2 \pi} x \sin x d x$$

Use a change of variables to evaluate the following definite integrals. $$\int_{0}^{1} 2 x\left(4-x^{2}\right) d x$$

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\pi / 4} \sec ^{2} \theta d \theta$$