91Ó°ÊÓ

Evaluate the integral. $$ \int \frac{d x}{x^{2} \sqrt{4 x^{2}-1}} $$

Find the volume obtained by rotating the region bounded by the curves about the given axis. $$ y=\sin x, y=0, \pi / 2 \leqslant x \leqslant \pi ; \quad \text { about the } x \text { -axis } $$

Use the method of cylindrical shells to ind the volume generated by rotating the region bounded by the curves about the given axis. \(y=\cos (\pi x / 2), y=0,0 \leqslant x \leqslant 1 ; \quad\) about the \(y\) -axis

Evaluate the integral. $$ \int \frac{d \theta}{1+\cos \theta} $$

(a) Show that \(\int_{-\infty}^{\infty} x d x\) is divergent. (b) Show that $$ \lim _{t \rightarrow \infty} \int_{-t}^{t} x d x=0 $$ This shows that we can't define $$ \int_{-\infty}^{\infty} f(x) d x=\lim _{t \rightarrow \infty} \int_{-t}^{t} f(x) d x $$

Evaluate the integral. $$ \int \frac{d \theta}{1+\cos ^{2} \theta} $$

Use the method of cylindrical shells to ind the volume generated by rotating the region bounded by the curves about the given axis. \(y=e^{x}, y=e^{-x}, x=1 ;\) about the \(y\) -axis

Find the volume obtained by rotating the region bounded by the curves about the given axis. $$ y=\sin ^{2} x, y=0,0 \leqslant x \leqslant \pi ; \text { about the } x \text { -axis } $$

Find the volume obtained by rotating the region bounded by the curves about the given axis. $$ y=\sin x, y=\cos x, 0 \leq x \leq \pi / 4 ; \quad \text { about } y=1 $$

Evaluate the integral. $$ \int \sqrt{x} e^{\sqrt{x}} d x $$