Exam 5: Integrals

The base of a solid is the region bounded by the graphs of $y = 4 - x ^ { 2 }$ and $y = 0$ . The cross sections perpendicular to the $y$ -axis are equilateral triangles. Find the volume of the solid.

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = 4 x , y = 4 x e ^ { 1 - x / 2 } , \text { about } y = 16$

Use a graphing utility to (a) plot the graphs of the given functions and (b) find the approximate x-coordinates of the points of intersection of the graphs.Then find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the x-axis. Round answers to two decimal places. $y = \frac { 1 } { 5 } x ^ { 5 } , y = 3 x ^ { 2 } - x ^ { 3 }$

Use the method of disks or washers, or 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. $y = 2 \sqrt { x } , y = x - 3 , y = 0$ , the $x$ -axis

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line.Sketch the region and a representative rectangle. $y = \sqrt { x - 1 } , y = x - 1 ; \text { the line } x = 2$

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

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y = x ^ { 2 } - 2 x - 1 , \quad y = - e ^ { x } - 1 , \quad x = 0 , \quad x = 1$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. $y = x ^ { 2 } , \mathrm { y } = 0 , x = 3 , x = 5 \text {; the } x \text {-axis }$

In a steam engine the pressure and volume of steam satisfy the equation $P V ^ { 1.4 } = k$ , where $k$ is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Calculate the work done by the engine (in ft-lb) during a cycle when the steam starts at a pressure of $160 \mathrm { lb } / \mathrm { in } ^ { 2 }$ and a volume of $100 \mathrm { in } ^ { 3 }$ and expands to a volume of $1000 \mathrm { in } ^ { 3 }$ . Use the fact that the work done by the gas when the volume expands from $V _ { 1 }$ to volume $V _ { 2 }$ is $W = \int _ { V _ { 1 } } ^ { V _ { 2 } } P d V$

Sketch the region enclosed by $y = 1 + \sqrt { x }$ and $y = \frac { 4 + x } { 4 }$ . Find the area of the region.

Use a graphing utility to (a) plot the graphs of the given functions and (b) find the x-coordinates of the points of intersection of the curves.Then find an approximation of the area of the region bounded by the curves using the integration capabilities of the graphing utility.Round answers to two decimal places. $y = 2 x ^ { 2 } , \mathrm { y } = 5 - x ^ { 4 }$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x$ -axis. $y = \cos x + 1 , x = 0 , y = 0 , x = \frac { 1 } { 2 } \pi$

Find the area of the region bounded by the curves. $x = 2 y ^ { 2 } , x = 16 + y ^ { 2 }$

Use the method of disks or washers, or 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. $y = 4 x ^ { 2 } , y = 8 x - 4 , y = 16 \text {; the } y \text {-axi }$

Find the average value of the function on the given interval. $g ( x ) = 5 x e ^ { - x ^ { 2 } } , [ 0,5 ]$

Newton's Law of Gravitation states that two bodies with masses $m _ { 1 }$ and $m _ { 2 }$ attract each other with a force $F = G \frac { m _ { 1 } m _ { 2 } } { r ^ { 2 } }$ where $r$ is the distance between the bodies and $G$ is the gravitation constant. If one of bodies is fixed, find the work needed to move the other from $r = l$ to $r = g$ .

Racing cars driven by Chris and Kelly are side by side at the start of a race.The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race.Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds. t 0 0 0 1 22 28 2 33 38 3 45 46 4 53 60 5 63 69 6 72 83 7 78 83 8 85 97 9 90 98 10 90 102

Use a graphing utility to (a) plot the graphs of the given functions and (b) find the approximate $x$ -coordinates of the points of intersection of the graphs. Then find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the $x$ -axis. Round answers to two decimal places. $y = \frac { 1 } { 5 } x ^ { 5 } , \mathrm { y } = 3 x ^ { 2 } - x ^ { 3 }$

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y = \sec ^ { 2 } x + 6 , y = 8 , x = - \frac { \pi } { 4 } , x = \frac { \pi } { 4 }$

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. $y = x ^ { 2 } , y = 0 , x = 3 \text {; the } y \text {-axis }$