Exam 5: Integrals

A particle moves a distance of $190 \mathrm { ft }$ along a straight line. As it moves, it is acted upon by a constant force of magnitude $15 \mathrm { lb }$ in a direction opposite to that of the motion. What is the work done by the force?

Find the area of the region bounded by the given curves. $y = \cos x , y = \sin 4 x , x = 0 , x = \frac { \pi } { 2 }$

$\text { Find the volume of a cap of a sphere with radius } r = 100 \text { and height } h = 6 \text {. }$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis.Sketch the region and a representative rectangle. $y = \frac { 1 } { x \left( x ^ { 2 } + 144 \right) } , \quad y = 0 , x = 1 , x = 12 \text {; the } y \text {-axis }$

Find the area of the region bounded by the curves. $y = \cos x , y = \sin 4 x , x = 0 , x = \frac { \pi } { 2 }$

Find the volume of the solid obtained by revolving the region under the graph of $y = 5 \left( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 4 } } \right)$ on $[ 1 , \infty )$ about the $x$ -axis.

Use cylindrical shells to find the volume of the solid. A sphere of radius $r$ .

Find the area of the shaded region.

The velocity $v$ of blood that flows in a blood vessel with radius $B$ and length $l$ at a distance $b$ from the central axis is $v ( r ) = \frac { P } { 4 q l } \left( B ^ { 2 } - b ^ { 2 } \right)$ , where $P$ is the pressure difference between the ends of the vessel and $q$ is the viscosity of the blood. Find the average velocity (with respect to $r$ ) over the interval $0 \leq b \leq B$ .

Find the area of the region bounded by the given curves (a) using integration with respect to x and (b) using integration with respect to y. $y = 2 x ^ { 3 } , y = 5 x + 6 , x = 0$

Find the volume of the frustum of a pyramid with square base of side $b = 17$ , square top of side $a = 5$ , and height $h = 19$ .

Find the area of the region bounded by the given curves. $y = x ^ { 2 } - 6 x , y = 5 x + 12$

Graph the region between the curves and use your calculator to compute the area correct to five decimal places. $y = 5 e ^ { 1 - x ^ { 2 } } , y = 5 x ^ { 4 }$

A chain weighing 4 lb/ft hangs vertically from a winch located 11 ft above the ground, and the free end of the chain is just touching the ground.Find the work done by the winch in pulling in the whole chain.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y = \sin x , y = 0 , x = 4 \pi , x = 11 \pi \text {; about the } y \text {-axis }$

If $4 \mathrm {~J}$ of work are needed to stretch a spring from $10 \mathrm {~cm}$ to $12 \mathrm {~cm}$ and another $20 \mathrm {~J}$ are needed to stretch it from $12 \mathrm {~cm}$ to $14 \mathrm {~cm}$ , what is the natural length of the spring? Round the answer to nearest integer.

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y = - x ^ { 2 } + 4 , y = 2 x + 1$

A tank has the shape of an inverted right circular cone with a base radius of $4 \mathrm {~m}$ and a height of $9 \mathrm {~m}$ . If the tank is filled to a height of $1 \mathrm {~m}$ , find the work required (to the nearest joule) to empty the tank by pumping the water over the top of the tank. (The mass of water is $1000 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ and the force of gravity is $9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ .)

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y = \sin 2 x , \mathrm { y } = 7 \cos x , x = \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 }$