Finding the Volume of a solid by integration

In summary, the conversation involves a student asking for help with two problems, one involving finding the volume of a solid obtained by rotating a region around the x-axis, and the other involving rotating a region around the y-axis. The student also mentions using the formula for volume of a solid of rotation and getting different answers from the expected ones. The expert summarizes the solution for both problems and clarifies that the first one has a volume of 9π while the second one has a volume of 12π. The expert also mentions that the problems should be posted in the "homework help" section and that the student should show their work before asking for help.
  • #1
calcstudent04
4
0
I need help with these 2 problems.

problem 1:
Find the volume if 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.

problem 2:
The regionbounded by the graph of f(x)=x^2+1 and the x-axis between x=0 and x=2 is rotted about the y-axis. Find the volume of the resulting soild
 
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  • #2
Have you covered solids of rotation in class?
 
  • #3
yes. Have you?
 
  • #4
Then you should know that if you revolve the area under the graph of f(x) for [tex]a \geq x \leq b[/tex] about the X axis, the volume is given by:
[tex]V = \pi \int _a^b f(x)^2dx[/tex]
 
  • #5
I worked the problem using that formula I just need to know if you got the same. For the first one I got 9pi and on the secong I got14pi
 
  • #6
I really would appreciate any help
 
  • #7
I got 6pi for the second one using cylindrical shells method

height of each cylinder is y = x^2 + 1

radius of each cylinder is simply "x"

circumference of each cylinder is 2pi times "x"

Then integrate over the interval [0,2]
 
  • #8
Sorry I meant to say second one is 12 pi
 
  • #9
1. These should be posted in the "homework help" section.

2. You should show us what you have tried on homework problems.

3. In the original post you did NOT ask us to verify your answer and did not tell us what you got.

4. The first figure is a cone with base radius and height both equal to 3 and so has volume 9π.

5. The second problem can be done by "shells" as jon said and the volume is indeed 12π.
 

1. What is the concept of finding the volume of a solid by integration?

The concept of finding the volume of a solid by integration is based on the fundamental principle of calculus, which states that the area under a curve can be determined by integrating the function that defines the curve. In this case, the curve represents the cross-sectional area of the solid, and by integrating it over the entire length of the solid, we can find its volume.

2. How is integration used to find the volume of a solid?

To find the volume of a solid using integration, we first need to find the function that represents the cross-sectional area of the solid. This can be done by slicing the solid into thin cross-sections and determining the area of each section. Then, we use integration to sum up all the infinitesimal areas and find the total volume of the solid.

3. Can integration be used to find the volume of any solid?

Yes, integration can be used to find the volume of any solid as long as its cross-sectional area can be represented by a function. This method is especially useful for finding the volumes of irregularly shaped solids, such as cones, spheres, and pyramids.

4. What are the benefits of using integration to find the volume of a solid?

The main benefit of using integration to find the volume of a solid is that it allows for the calculation of volumes of complex shapes that cannot be easily measured or calculated using traditional methods. Additionally, integration is a more accurate method of finding volume compared to approximations or estimations.

5. Are there any limitations to using integration to find the volume of a solid?

One limitation of using integration to find the volume of a solid is that it requires a good understanding of calculus and mathematical concepts. This method may also be time-consuming and tedious for more complex shapes. Additionally, it may not be practical to use integration for finding the volume of very large or irregularly shaped solids.

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