Exam 6: Applications of Integration

A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, find the hydrostatic force in pounds on the side of the tank.

Find the area of the region bounded by the curves $x = y ^ { 2 } - 7$ and $x = y - 1$ .

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

Find the volume of the solid obtained when the region bounded by the curve $y = \sin x , 0 \leq x \leq \pi$ and the x-axis is rotated about the y-axis.

The area of the region bounded by $y ^ { 2 } = x , ( y - 2 ) ^ { 2 } = x$ and the y-axis is

Express the area of the given region as a definite integral. Do not evaluate.

Find the area of the region bounded by the curves $f ( x ) = 3 \cdot \sin \left( \frac { \pi } { 4 } x \right) , g ( x ) = 2 ^ { x } - 1 \text { and } x \geq 0$

Two people weighing 100 lb and 180 lb are at opposite ends of a seesaw 14 ft long. Where should the point of support be placed so that the seesaw will balance?

Find c so that the following can serve as the probability density function of a random variable X: $f ( X ) = \left\{ \begin{array} { l l } c \cdot X \cdot e ^ { - 4 \mathrm { x } ^ { 2 } } & \text { if } \mathrm { X } \geq 0 \\0 & \text { otherwise }\end{array} \right.$

Find the volume of the solid obtained by rotating the region bounded by $y = 2 x - x ^ { 2 }$ and the x-axis about the line x = 2.

A particle is moving along a straight line so that its velocity at time t is v(t) = 3t². At what time t during the interval $0 < t < 3$ is its velocity the same as its average velocity over the entire interval?

The marginal cost for a company is given by $C ^ { \prime } ( q ) = 2 - 0.01 q + 0.005 q ^ { 2 }$ where q is the number of units produced. What is the total cost to raise production from 100 to 200 units?

Find the volume of the solid generated by rotating about the line $y = - 1$ the region bounded by the graphs of the equations $y = x ^ { 2 } - 4 x + 5 \text { and } y = 5 - x$

Find the length of the curve $x = t ^ { 3 } \cos t , y = t ^ { 3 } \sin t , 0 \leq t \leq 3$ using a graphing calculator to evaluate the integral.

Find the arc length of the curve $y = x ^ { 2 } - \frac { 1 } { 8 } \ln x , \quad 1 \leq x \leq 3$ .

Find the area of the region bounded by $y = x ^ { 3 } - 6 x \text { and } y = - 2 x$

Find the area of the region bounded by the curves $y = x ^ { 2 } + 1$ and $y = 2$

Estimate the average value of the function whose graph is given below.

Let X be a continuous random variable with density function $f ( X ) = \left\{ \begin{array} { l l } c \cdot e ^ { - c X } & \text { if } x \geq 0 \\0 & \text { otherwise }\end{array} \right.$ If the median of this distribution is $\frac { 1 } { 3 }$ , then c is:

Find the arc length of the curve $3 y = 4 x$ from (3, 4) to (9, 12).