Exam 9: Techniques of Integration

Solve the problem. -An oil storage tank can be described as the volume generated by revolving the area bounded by $y = \frac { 28 } { \sqrt { 49 + x ^ { 2 } } }$ , $x = 0 , y = 0 , x = 2$ about the $x$ -axis. Find the volume (in $\mathrm { m } ^ { 3 }$ ) of the tank.

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 5 } ^ { 6 } \left( 4 x ^ { 3 } + 1 x \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.

Determine whether the improper integral converges or diverges. - $\int _ { 4 } ^ { \infty } \frac { d x } { x ^ { 7 / 2 } }$

Use reduction formulas to evaluate the integral. - $\int \csc ^ { 4 } \frac { t } { 5 } d t$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int _ { 3 } ^ { 7 } \frac { 4 x d x } { ( x - 2 ) ^ { 3 } }$

Find the area or volume. -Find the volume of the solid generated by revolving the region in the first quadrant under the curve $y = \frac { 4 } { x ^ { 2 } }$ , bounded on the left by $x = 1$ , about the $x$ -axis.

Use various trigonometric identities to simplify the expression then integrate. - $\int \sin \theta \cos \theta \cos 4 \theta d \theta$

Evaluate the integral. - $\int _ { 0 } ^ { \infty } e ^ { - x } \cos 4 x d x$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 4 x ^ { 3 } + 20 x ^ { 2 } + 32 x + 13 } { x ^ { 2 } \left( x ^ { 2 } + 6 x + 13 \right) } d x$

Use a trigonometric substitution to evaluate the integral. - $\int _ { 0 } ^ { 1 } \frac { e ^ { x } d x } { 4 - e ^ { 2 x } }$

Evaluate the integral. - $\int \frac { d x } { x ^ { 2 } + 6 x + 25 }$

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 Simpson's Rule. Width (ft) Depth (ft) 0 4 3 4.5 6 5 9 6 12 6.5 15 7 18 8

Evaluate the integral. - $\int \sec ^ { 3 } 9 x d x$

Evaluate the integral. - $\int _ { 0 } ^ { 1 / 2 } 3 \sin ^ { 4 } 2 \pi x d x$

Solve the problem. -There are 3 balls in a hat; one with the number 1 on it, one with the number 5 on it, and one with the number 9 on it. You pick a ball from the hat at random and then you flip a coin to obtain heads (H) or tails (T). Determine the set of possible outcomes, then find the probability that the number on the ball is greater than $6 .$

Evaluate the integral. - $\int \frac { d x } { \left( 16 - x ^ { 2 } \right) ^ { 2 } }$

Evaluate the integral. - $\int _ { 0 } ^ { \pi } \sin 7 t \cos 6 t d t$

Solve the problem by integration. - $\text { Find the area bounded by } y = \frac { x - 14 } { x ^ { 2 } - 6 x - 40 } , y = 0 , x = 2 \text {, and } x = 4 \text {. Round to the nearest hundredth. }$

Evaluate the integral by using a substitution prior to integration by parts. - $\int \cos ( \ln x ) d x$

Solve the initial value problem for x as a function of t. - $( t + 4 ) \frac { d x } { d t } = x ^ { 2 } + 1 , \quad t > - 4 , x ( 1 ) = \tan 1$