91Ó°ÊÓ

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 \(x=\pi / 2\).

In Exercises \(29-36\) , use an appropriate substitution and then a trigonometric substitution to evaluate the integrals. $$ \int \frac{x d x}{\sqrt{x^{2}-1}} $$

Evaluate the integrals in Exercises \(33-38\). $$ \int_{-\pi}^{\pi} \sin 3 x \sin 3 x d x $$

Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int_{1}^{e^{\pi / 3}} \frac{d x}{x \cos (\ln x)} $$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int 8 \sin 4 t \sin \frac{t}{2} d t\)

In Exercises \(35-64\) , 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. $$ \int_{0}^{\pi / 2} \tan \theta d \theta $$

Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int \frac{\ln x d x}{x+4 x \ln ^{2} x} $$

Evaluate the integrals in Exercises \(33-38\). $$ \int_{0}^{\pi / 2} \sin x \cos x d x $$

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 \(x=\pi\) (See Exercise 31 for a graph.)

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \sin \frac{t}{3} \sin \frac{t}{6} d t\)