91Ó°ÊÓ

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{\pi}^{\infty} \frac{1+\sin x}{x^{2}} d x$$

Evaluate the integrals. $$\int_{-\pi / 2}^{\pi / 2} \cos x \cos 7 x d x$$

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve \(y=e^{-x},\) and the line \(x=1\) a. about the \(y\) -axis. b. about the line \(x=1\).

Surface area Find the area of the surface generated by revolving the curve \(y=\sqrt{x^{2}+2}, 0 \leq x \leq \sqrt{2},\) about the \(x\) -axis.

Require the use of various trigonometric identities before you evaluate the integrals. $$\int \sin ^{2} \theta \cos 3 \theta d \theta$$

Find, to two decimal places, the \(x\) -coordinate of the centroid of the region in the first quadrant bounded by the \(x\) -axis, the curve \(y=\tan ^{-1} x,\) and the line \(x=\sqrt{3}\).

Evaluate \(\int x^{3} \sqrt{1-x^{2}} d x\) using a. integration by parts. b. a \(u\) -substitution. c. a trigonometric substitution.

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve \(y=\cos x, 0 \leq x \leq \pi / 2,\) about a. the \(y\) -axis. b. the line \(x=\pi / 2\)

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{4}^{\infty} \frac{2 d t}{t^{3 / 2}-1}$$

Find the volume of the solid generated by revolving the region bounded by the \(x\) -axis and the curve \(y=x \sin x, 0 \leq x \leq \pi,\) about a. the y-axis. b. the line \(x=\pi\).