Exam 6: Additional Topics in Integration

Evaluate $\int x e ^ { x / 8 } d x$ .

Evaluate $\int x e ^ { x / 3 } d x$ .

Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 30 } } d x$

Determine how many subintervals are required to guarantee accuracy to within 0.00001 for the approximation of the integral $\int _ { 1 } ^ { 2 } \ln \left( 1 + x ^ { 2 } \right) d x$ using (a) the trapezoidal rule and (b) Simpson's rule.

A hospital patient receives intravenously 16 units of a certain drug per hour. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 11 }$ . If the treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run? Round to two decimal places, if necessary.

Evaluate $\int x ^ { 7 } \left( x ^ { 4 } - 10 \right) ^ { 7 } d x$ .

Evaluate $\int x e ^ { x / 6 } d x$ .

Given an initial population, $P _ { 0 } = 17,000$ , a renewal rate, R = 150, and a survival function, $S ( t ) = t e ^ { - 0.2 t }$ , with time t measured in years, determine the population at the end of 11 years. Round to two decimal places.

Evaluate $\int x ( x + 2 ) ^ { 7 } d x$ .

Evaluate $\int x ( x + 7 ) ^ { 8 } d x$ .

Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 11 } } d x$ .

The life span of car stereos manufactured by a certain company is measured by a random variable X that is exponentially distributed with a probability density function $f ( x ) = \left\{ \begin{array} { l l } 0.2 e ^ { - 0.2 x } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ where x is the life span in years of a randomly selected stereo. What is the probability that the life span of a randomly selected stereo is between 1 and 15 years? Round to the nearest hundredth.

A hospital patient receives intravenously 6 units of a certain drug per hour. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 11 }$ . If the treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run? Round to two decimal places, if necessary.

After t hours on the job, a factory worker can produce $130 t e ^ { - 0.5 t }$ units per hour. How many units does the worker produce during the first 4 hours? Round to two decimal places.

A 2.5-hour movie runs continuously at a local theater. You leave for the theater without first checking the show times. Use an appropriate uniform density function to find the probability that you will arrive at the theater within 4 minutes of the start of the film (before or after). Round to the nearest hundredth.

$f ( x ) = \left\{ \begin{array} { l r } \frac { 1 } { 7 } & \text { for } 0 \leq x \leq 7 \\0 & \text { otherwise }\end{array} \right.$ is a probability density function for a particular random variable X. Use integration to find $P ( 0 \leq x \leq 3 )$ rounded to the nearest hundredth.

A hospital patient receives intravenously 17 units of a certain drug per hour. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 14 }$ . If the treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run? Round to two decimal places, if necessary.

Let X be a random variable that measures the age of a randomly selected virus in a particular population. Suppose X is exponentially distributed with a probability density function $f ( x ) = \left\{ \begin{array} { l l } k e ^ { - k x } & \text { for } x \geq 0 \\0 & \text { otherw ise }\end{array} \right.$ where x is the age of a randomly selected virus and k is a positive constant. Experiments indicate that it is four times as likely for a virus to be less than 2 days old as it is for it to be more than 2 days old. Use this information to determine k.