Exam 6: Additional Topics in Integration

The time interval between the arrivals of successive trains at a certain station is measured by a random variable X with a probability density function $f ( x ) = \left\{ \begin{array} { l l } 0.35 e ^ { - 0.35 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ where x is the time (in minutes) between the arrivals of a randomly selected pair of successive trains. What is the probability that two successive trains selected at random will arrive within 4 minutes of one another? Round to the nearest hundredth.

Suppose the length of time, x, that it takes a chimpanzee to solve a simple puzzle is measured by a random variable X that is exponentially distributed with a probability density function $f ( x ) = \left\{ \begin{array} { l l } \frac { 2 } { 3 } e ^ { - 2 x / 3 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ where x is in minutes. Find the probability that a randomly chosen chimpanzee will take more than 12 minutes to solve the puzzle.

Use integration by parts to evaluate the integral $\int ( 1 - x ) e ^ { x } d x$

Given an initial population, $P _ { 0 } = 14,000$ , a renewal rate, R = 100, 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.

$\int _ { 0 } ^ { \infty } \frac { 1 } { ( 1 + x ) ^ { 3 } } d x = \frac { 1 } { 8 }$ .

Evaluate $\int _ { 2 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x ( x - 1 ) } }$ .

Assume a 7-year franchise is expected to generate profit at the rate of $\sqrt [ 3 ] { t ^ { 2 } + 12,000 }$ dollars per year. If, over the next 7 years, the prevailing annual interest rate remains fixed at 4%, compounded continuously, what is the present value of the franchise? Use Simpson's rule with n = 6 to approximate the integral. Round your answer to two decimal places.

An object moving in a straight line has velocity $v ( t ) = t e ^ { - \sqrt { t } }$ meters per second. Is it true that in the first 4 seconds the object will have travelled $4 e ^ { 2 } + 12$ meters?

Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 8 } } d x$ .

It is projected that t years from now the population of a city will be changing at the rate of $t ^ { 2 } e ^ { 0.01 t }$ thousand people per year. If the current population is 1 million, what will the population be 4 years from now?

Evaluate the improper integral: $\int _ { 1 } ^ { \infty } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 6 \right) ^ { 2 } } d x$

The time interval between the arrivals of successive trains at a certain station is measured by a random variable X with a probability density function $f ( x ) = \left\{ \begin{array} { l l } 0.4 e ^ { - 0.4 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ where x is the time (in minutes) between the arrivals of a randomly selected pair of successive trains. What is the probability that two successive trains selected at random will arrive within 2 minutes of one another? Round to the nearest hundredth.

From time t = 0 to t = 5 an object's speed is given by the function $s ( t ) = t e ^ { 4 t }$ . Compute the distance travelled by the object during this time interval. Round your answer to two decimals.

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 8 x } d x$

The useful life X of a particular kind of machine is a random variable with density function $f ( x ) = \left\{ \begin{array} { l l } \frac { 4 } { 15 } + \frac { 2 } { 3 x ^ { 2 } } & \text { if } 2 \leq x \leq 5 \\0 & \text { otherwise }\end{array} \right.$ where x is the number of years a randomly selected machine stays in use. $P ( X \leq 3 ) = \frac { 28 } { 45 }$

Evaluate the improper integral: $\int _ { 2 } ^ { \infty } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 6 \right) ^ { 2 } } d x$

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 5 x } d x$ .

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 5 } / 6 } d x$ Round to two decimal places, if necessary.

Evaluate $\int x \ln \sqrt { x } d x$ .

Use the formula $\int ( \ln x ) ^ { n } d x = x ( \ln x ) ^ { n } - n \int ( \ln x ) ^ { n - 1 } d x$ to evaluate $\int ( \ln x ) ^ { 3 } d x$ .