Exam 6: Applications of Integration

The density function for the waiting time at a bank is modeled by $p ( t ) = 0.1 e ^ { - 0.1 t } , t > 0$ and is measured in minutes.(a) Find the median waiting time.(b) Find the mean waiting time.(c) Sketch the graph of the density function showing the median and mean.

Find the area of the region bounded by the curve $x = 4 \cos t , y = 3 \sin t , 0 \leq t \leq \pi$ , and the x-axis.

Find the arc length of the curve $y ^ { 2 } = x ^ { 3 }$ from (0, 0) to ( $\frac { 1 } { 4 }$ , $\frac { 1 } { 8 }$ ).

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.

Let $f ( x ) = 8 x e ^ { - 4 x ^ { 2 } } , x > 0$ be the probability density function of a random variable X. Find the median of the probability density function $f$ .

Find the length of the curve $y = \frac { 2 } { 3 } ( x - 4 ) ^ { \frac { 3 } { 2 } } , 7 \leq x \leq 12$ .

Find the volume of the solid obtained when the region bounded by the x-axis, the y-axis, and the line $y + x = 3$ is rotated about the x-axis.

Consider the region R bounded by $y = \frac { 1 } { 2 } \sqrt { x } , y = 1$ , and the y-axis.(a) Find the area of R.(b) Find the average height of R.(c) Find the volume, V, of the solid obtained by rotating R about the x-axis.(d) A cross section of the solid generated by part (c) taken perpendicular to the x-axis is a washer. Determine the average area of the cross sections of the solid.

If $\frac { 15 } { 4 }$ ft-lb of work is needed to stretch a spring a length $\frac { 1 } { 2 }$ ft beyond its natural length, find the spring constant.

Find the x-coordinate of the centroid of the region bounded by the graphs $y = \sqrt [ 3 ] { x } , \quad x = 8$ and the x-axis.

Let $f ( x ) = ( 3 - x ) ( 3 + x )$ , find c such that $f _ { a v e } = f ( c )$ on the interval $[ 0,3 ]$ .

Find the volume of the solid obtained when the region bounded by the curves $y = x ^ { 3 }$ , $x = 1$ , and the x-axis is rotated about the line x = - 1.

Find the area of the region bounded by the curves $y = x ^ { 3 } + x ^ { 2 }$ and $y = 2 x ^ { 2 } + 2 x$ .

Let $f ( t ) = 0.2 e ^ { - 2 t } , t \geq 0$ be the probability density function of a random variable T, where t is the time that a customer spends in line at teller's window before being served. What is the probability that a customer will wait more than 10 minutes?

The area of the region bounded by $y = \sin x$ and $y = \cos x$ between $x = \frac { \pi } { 4 }$ and $x = \frac { 5 \pi } { 4 }$ is

Suppose a company has estimated that the marginal cost of manufacturing x pairs of a new line of jeans is $c ^ { \prime } ( x ) = 3 + 0.002 x + 0.000006 x ^ { 2 }$ (measured in dollars per pair) with a fixed start-up cost of c(0) = 2000. Find the cost of producing the first 1000 pair of jeans.

Consider the region in the xy-plane between $x = 0 , x = \frac { \pi } { 2 } , \text { bounded by } y = 0$ and $y = \sin x$ . Find the volume of the solid generated by rotating this region about the x-axis.

(a) Explain why the function defined by the graph below is a probability density function. (b) Use the graph to find the following probabilities: (i) $P ( x < 2 )$ (ii) $P ( 2 \leq x \leq 5 )$ (c) Calculate the median for this distribution.

The voltage (in volts) at an electrical outlet is a function of time (in seconds) given by V(t) = V₀ cos(120 $\pi$ ) where V₀ is a constant representing the maximum voltage.(a) What is the average value of the voltage over one second? (b) How many times does the voltage reach a maximum in one second? (c) Define the new function $S ( t ) = ( V ( t ) ) ^ { 2 }$ . Compute $\overline{S}$ , the average value of $S ( t )$ over one cycle.(d) Instead of the average voltage, engineers use the root mean square $V _ { n m s } = \sqrt { \bar { S } }$ . Determine V_rms in terms of V₀.(e) The standard household voltage in the United Stated is 100 volts. This means that V_rms = 110. What is the value of V₀?

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