Define
as the region bounded above by the graph of
and below by the
over the interval
Find the volume of the solid of revolution formed by revolving
around the line
For our final example in this section, let’s look at the volume of a solid of revolution for which the region of revolution is bounded by the graphs of two functions.
A region of revolution bounded by the graphs of two functions
Define
as the region bounded above by the graph of the function
and below by the graph of the function
over the interval
Find the volume of the solid of revolution generated by revolving
around the
First, graph the region
and the associated solid of revolution, as shown in the following figure.
Note that the axis of revolution is the
so the radius of a shell is given simply by
We don’t need to make any adjustments to the
x -term of our integrand. The height of a shell, though, is given by
so in this case we need to adjust the
term of the integrand. Then the volume of the solid is given by
Define
as the region bounded above by the graph of
and below by the graph of
over the interval
Find the volume of the solid of revolution formed by revolving
around the
We have studied several methods for finding the volume of a solid of revolution, but how do we know which method to use? It often comes down to a choice of which integral is easiest to evaluate.
[link] describes the different approaches for solids of revolution around the
It’s up to you to develop the analogous table for solids of revolution around the
Let’s take a look at a couple of additional problems and decide on the best approach to take for solving them.
Selecting the best method
For each of the following problems, select the best method to find the volume of a solid of revolution generated by revolving the given region around the
and set up the integral to find the volume (do not evaluate the integral).
The region bounded by the graphs of
and the
The region bounded by the graphs of
and the
First, sketch the region and the solid of revolution as shown.
Looking at the region, if we want to integrate with respect to
we would have to break the integral into two pieces, because we have different functions bounding the region over
and
In this case, using the disk method, we would have
If we used the shell method instead, we would use functions of
to represent the curves, producing
Neither of these integrals is particularly onerous, but since the shell method requires only one integral, and the integrand requires less simplification, we should probably go with the shell method in this case.
First, sketch the region and the solid of revolution as shown.
Looking at the region, it would be problematic to define a horizontal rectangle; the region is bounded on the left and right by the same function. Therefore, we can dismiss the method of shells. The solid has no cavity in the middle, so we can use the method of disks. Then
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?