Exam 6: Applications of Integration

Find the volume of the solid obtained by rotating the region bounded by the curves $y = 2 x ^ { 2 } , x = 0 , \text { and } y = 2$ about the y-axis.

(a) Show that $p ( x ) = \left\{ \begin{array} { l l } ( 2 - k ) x ^ { - 3 + k } & \text { if } x \geq 1 \\0 & \text { otherwise }\end{array} \right.$ where k is fixed and $0 < k < 1$ is a probability density function.(b) What is the mean for this distribution? (c) Calculate the median of $p$

The base of a solid $S$ is the parabolic region $\left\{ ( x , y ) \mid y ^ { 2 } \leq x \leq 1 \right\}$ . Cross-sections perpendicular to the x-axis are squares. Find the volume of $S$ .

Find the average value of the function $f ( x ) = \sin 3 x \text { on the interval } \left[ 0 , \frac { \pi } { 3 } \right]$

Find the volume of the solid obtained by rotating the region bounded by the curves $y = \sqrt { x - 2 } , y = 0 , \text { and } x = 6$ about the x-axis.

Suppose that the mileage (in thousands of miles) which car owners can obtain from a certain kind of tire has the probability density $p ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 100 } \cdot e ^ { - \frac { x } { 100 } } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ Find the probability that a tire chosen at random will last (a) at most 10,000 miles.(b) between 15,000 and 25,000 miles.(c) at least 30,000 miles.

The density function for the life of a certain type of battery is modeled by $p ( t ) = 0.2 e ^ { - 0.2 t } , t > 0$ and is measured in months.(a) What is the probability that a battery will wear out during the first month of use? (b) What is the probability that a battery is functioning for more than 12 months?

Find the area of the region bounded by the curves $y = \ln x , x = e$ and the x-axis.

The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by $c ( t ) = \frac { 1 } { 15 } t ( 15 - t ) , \quad 0 \leq t \leq 15$ , where t is measured in seconds. Find the cardiac output.