91Ó°ÊÓ

In Exercises \(35-68\) , 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. $$\int_{\pi}^{\infty} \frac{2+\cos x}{x} d x$$

Finding volume 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 \(x=1\)

In Exercises \(35-68\) , 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. $$\int_{\pi}^{\infty} \frac{1+\sin x}{x^{2}} d x$$

Consider the region bounded by the graphs of \(y=\sqrt{x \tan ^{-1} x}\) and \(y=0\) for \(0 \leq x \leq 1 .\) Find the volume of the solid formed by revolving this region about the \(x\) -axis (see accompanying figure).

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

Moment about \(y\) -axis A thin plate of constant density \(\delta=1\) occupies the region enclosed by the curve \(y=36 /(2 x+3)\) and the line \(x=3\) in the first quadrant. Find the moment of the plate about the \(y\) -axis.

Use any method to evaluate the integrals in Exercises \(55-66\) $$ \int \frac{x^{4}-1}{x^{5}-5 x+1} d x $$

Use any method to evaluate the integrals in Exercises \(55-66\) $$ \int \frac{\ln x+2}{x(\ln x+1)(\ln x+3)} d x $$

Finding volume 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 \(x=\pi / 2\)

Require the use of various trigonometric identities before you evaluate the integrals. \(\int \sin \theta \cos \theta \cos 3 \theta d \theta\)