Exam 9: Techniques of Integration

Find the indicated probability. - $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 4 } \mathrm { e } ^ { - \mathrm { x } / 4 } ; [ 0 , \infty ) , \mathrm { P } ( 1 \leq \mathrm { x } \leq 4 )$

Determine whether the function is a probability density function over the given interval. - $f ( x ) = 5 \sin 5 x \text { over } \left[ \frac { \pi } { 10 } , \frac { \pi } { 5 } \right]$

Determine whether the improper integral converges or diverges. - $\int _ { 4 } ^ { 5 } \frac { e ^ { - \sqrt { x - 4 } } } { \sqrt { x - 4 } } d x$

Evaluate the integral. - $\int x ^ { 2 } \ln 9 x d x$

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { \sqrt { 6 x + 2 } } { x ^ { 2 } }$

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } } d x$

Solve the problem. -Find an upper bound for $| \mathrm { E } _S |$ in estimating $\int _ { 0 } ^ { 5 } \left( 6 \mathrm { x } ^ { 2 } - 8 \right) \mathrm { dx }$ with $\mathrm { n } = 8$ steps.

Evaluate the integral. - $\int \frac { x + 10 } { x + 2 } d x$

Evaluate the integral. - $\int \frac { 1 } { x \sqrt { 49 + x ^ { 2 } } } d x$

Use a trigonometric substitution to evaluate the integral. - $\int \frac { d x } { x \left( 1 + 25 \ln ^ { 2 } x \right) }$

Expand the quotient by partial fractions. - $\frac { 5 x + 3 } { ( x - 3 ) ( x - 1 ) }$

Integrate the function. - $\int \sqrt { 16 - x ^ { 2 } } d x$

Evaluate the integral. - $\int y ^ { 3 } e ^ { - 7 y } d y$

Find the area or volume. -Find the volume of the solid generated by revolving the region under the curve $\mathrm { y } = 8 \mathrm { e } ^ { - 2 \mathrm { x } }$ in the first quadrant about the $y$ -axis.

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - $\int _ { 0 } ^ { 9 } x d x$

Solve the problem. -A rectangular swimming pool is being constructed, 18 feet long and 100 feet wide. The depth of the pool is measured at 3 -foot intervals across the length of the pool. Estimate the volume of water in the pool using the Trapezoidal Rule. Width (ft) Depth (ft) 0 5 3 5.5 6 6 9 7 12 7.5 15 8 18 9

Solve the problem. -The following table shows the rate of water flow (in gal/min) from a stream into a pond during a 30 -minute period after a thunderstorm. Use Simpson's Rule to estimate the total amount of water flowing into the pond during this period. Round your answer to the nearest gallon. Time (min) Rate (gal/min) 0 200 5 250 10 300 15 250 20 220 25 200 30 150

Solve the problem. -Find the length of the curve $y = \sqrt { 36 - x ^ { 2 } }$ between $x = 0$ and $x = 3$ .

Find the area or volume. -Find the volume of the solid generated by revolving the region under the curve $y = \frac { 7 } { x }$ , from $x = 1$ to $x = \infty$ , about the $x$ -axis.

Integrate the function. - $\int \frac { 9 \mathrm { dx } } { \sqrt { 9 + \mathrm { x } ^ { 2 } } }$