Exam 5: Integrals

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. $y = 16 - x ^ { 2 } , y = 0 ;$ the line $x = - 4$

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. $y ^ { 2 } - 3 y + x = 0 , x = 0$

The base of a solid is a circular disk with radius .Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the Base.

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the specified axis. $y = x ^ { 2 } , y = 0 , x = 1 , x = 6 ;$ about $x = 1$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. $y = 4 - x ^ { 2 } , y = 0 ;$ the line $y = 6$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. $y = \frac { e ^ { x / 2 } } { \left( 1 + e ^ { x } \right) ^ { 5 / 2 } } , \quad y = 0 , \quad x = 0 , x = 4 \text {; the } x \text {-axis }$

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 $W = \int _ { V _ { 1 } } ^ { V _ { 2 } } P d V$

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. $y = \sqrt { x - 1 } , y = x - 1 ; \text { the line } x = 2$

Find the area of the region bounded by the graph of $f ( x ) = \sqrt { x }$ , the $y$ -axis, and the tangent line to the graph of $f$ at $( 9,3 )$ .

A bucket weighs $5 \mathrm { lb }$ and a rope of negligible weight are used to draw water from a well that is 40 ft deep. The bucket starts with $50 \mathrm { lb }$ of water and is pulled up at a rate of $10 \mathrm { ft } / \mathrm { s }$ , but water leaks out of a hole in the bucket at a rate of $0.5 \mathrm { lb } / \mathrm { s }$ . Find the work done in pulling the bucket to the top of the well.

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

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

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. $y = 2 x ^ { 3 } , y = 5 x + 6 , x = 0$

The base of $S$ is a circular region with boundary curve $36 x ^ { 2 } + 16 y ^ { 2 } = 144$ . Cross-sections perpendicular to the $x$ axis are isosceles right triangles with hypotenuse in the base. Find the volume of $S$ .

The tank shown is full of water. Given that water weighs $62.5 \mathrm { lb } / \mathrm { ft }$ and $\mathrm { R } = 5$ , find the work (in $\mathrm { lb }$ -ft) required to pump the water out of the tank.

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.

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.

A heavy rope, $40 \mathrm { ft }$ long, weighs $0.7 \mathrm { lb } / \mathrm { ft }$ and hangs over the edge of a building $110 \mathrm { ft }$ high. How much work is done in pulling the rope to the top of the building?

Find the area of the region bounded by the parabola $y = x ^ { 2 }$ , the tangent line to this parabola at $( 3,9 )$ , and the $x$ axis.

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height $t$ as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius $d$ through the center of a sphere of radius $D$ and express the answer in terms of $t$ .