91Ó°ÊÓ

Evaluate the integral. \( \displaystyle \int \frac{dx}{x^2 \sqrt{4x^2 - 1}} \)

Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct \( \displaystyle \int_0^2 \sin 2 \pi x \cos 5 \pi x dx \)

(a) Evaluate the integral \( \displaystyle \int_0^\infty e^{-x}\ dx \) for \( n = 0, 1, 2 \), and \( 3 \). (b) Guess the value of \( \displaystyle \int_0^\infty x^n e^{-x}\ dx \) when \( n \) is an arbitrary positive integer. (c) Prove your guess using mathematical induction.

(a) Show that \( \displaystyle \int_{-\infty}^\infty x\ dx \) is divergent. (b) Show that $$ \lim_{t\to\infty} \int_{-t}^t x\ dx = 0 $$ This shows that we can't define $$ \int_{-\infty}^\infty f(x)\ dx = \lim_{t\to\infty} \int_{-t}^t f(x)\ dx $$

Evaluate the integral. \( \displaystyle \int \frac{d \theta}{1 + \cos \theta} \)

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. \( y = \cos (\frac{\pi x}{2}) \) , \( y = 0 \) , \( 0 \le x \le 1 \) ; about the y-axis

Find the volume obtained by rotating the region bounded by the curves about the given axis. \( y = \sin x \) , \( y = 0 \) , \( \frac{\pi}{2} \le x \le \pi \) ; about the x-axis

The average speed of molecules in an ideal gas is $$ \overline{v} = \frac{4}{\sqrt{\pi}} \left (\frac{M}{2RT} \right)^{\frac{3}{2}} \int_0^\infty v^3 e^{\frac{-Mv^2}{(2RT)}}\ dv $$ where \( M \) is the molecular weight of the gas, \( R \) is the gas constant, \( T \) is the gas temperature, and \( v \) is the molecular speed. Show that $$ \overline{v} = \sqrt{\frac{8RT}{\pi M}} $$

Find the volume obtained by rotating the region bounded by the curves about the given axis. \( y = \sin^2 x \) , \( y = 0 \) , \( 0 \le x \le \pi \) ; about the x-axis

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