Area and Volume (shell and disk method)

In summary, you are struggling with some problems related to finding volumes of solids using different methods such as cylindrical shells and circular disks. You have provided answers for some of the problems but are unsure if they are correct. It is recommended to focus on one problem at a time and seek help on a specific problem rather than posting a large set of questions.
  • #1
myanmar
33
0
Last few I can't seem to do, but on the others, can someone check my answers?. The last few I need help getting through.

1. Find the volume of the solid formed when the region bounded by the curves y = x^3 + 1, x = 1 and y = 0 is rotated about the x-axis.
My answer: int(pi(x^3+1)^2,x,-1,1)

2. Find the volume of the solid of revolution obtained by revolving the region bounded by y = 1/x and the lines x = pi/8 and x = pi/2 around the x-axis the x-axis.
My answer: int(pi(1/x)^2,x,pi/2,pi/8)

3. Find the area bounded by the curves f(x) = x^3 + x^2 and g(x) = 2x^2 + 2x.
int((x^3+1)-(2x^2+2x),x,-1,0) + int((2x^2+2x)-(x^3+1),x,0,2)

4. Find the area bounded by the curves y = sqrt(x), y = (5-x)/4, and y = (3x-8)/2
int(sqrt(x)-(5x-4)/4,x,1,3) + int(sqrt(x) - (3x-8)/2,x,3,4)

5. Let R be the region bounded by: y = x^3, the tangent to y = x^3 at (1,1), and the x-axis. Find the area of R integrating: a) with respect to x, and b) with respect to y.
Solving for tangent yields tangent = 3x-2
integrating for x yields
int(x^3-(3x-2),x,0,1)
b) Help me with this, not sure how to do at all.

6. Find the volume of the solid generated by revolving the region bounded by the graphs of y = x^2 - 4x + 5 and y = 5 - x about the line y = -1.
disk: int(pi((5-x)^2-(x^2-4x+5)^2),x,0,2)
shell: int(2pix(-x^2+3x),0,3)

7. Use cylindrical shells to find the volume of the solid obtained by revolving around the y-axis the region bounded by the curves y^2 = 8x and x = 2.
int(2pix(2sqrt(2x),x,0,2)

8. A cylindrical hole is drilled through the center of a sphere of radius R. Use the method of cylindrical shells to find the volume of the remaining solid, given that the solid is 6 cm high.
I believe the answer is 36 pi in^3, but can not seem to do this problem with the shell method.

9. Find the volume of the solid generated by revolving about the line x = -1, the region bounded by the curves y = -x^2 + 4x - 3 and y = 0.

10. Consider the region in the xy-plane between x = 0 and x=pi/2 bounded by y = 0 and y = sin x. Find the volume of the solid generated by revolving this region about the x-axis.

11. Let R be the region bounded by y = 1/x, y = x^2, x = 0, and y = 2. Suppose R is revolved around the x-axis. Set up but do not evaluate the integrals for the volume of rotation using: a) the method of cylindrical shells; b) the method of circular disks.
 
Physics news on Phys.org
  • #2
Posting 11 questions at once (including 'check my work' problems) really isn't going to help you get a response. People who answer these want to focus on what your problem is, not the whole problem set. My eyes glazed over when I saw this. That said, the first one looks ok. Now I'll skip ahead to where you actually ask a question. Part 5a) isn't so good. The tangent line hits the x-axis at x=2/3. You can't do this one dx in a single integral, can you?
 
  • #3
Since I can't seem to find the edit button, I'll post only what I need here. Ignore other problems

So, I'd appreciate some help on the following. Even one problem would be great.

In order of preference:
5, 10, 11, 9. (6&7 if possible)

5. (I figured out how to do b, but if #5a is wrong, I don't know where I went wrong.)

9. Find the volume of the solid generated by revolving about the line x = -1, the region bounded by the curves y = -x^2 + 4x - 3 and y = 0.
a little stuck

10. Consider the region in the xy-plane between x = 0 and x=pi/2 bounded by y = 0 and y = sin x. Find the volume of the solid generated by revolving this region about the x-axis.
stuck here too

11. Let R be the region bounded by y = 1/x, y = x^2, x = 0, and y = 2. Suppose R is revolved around the x-axis. Set up but do not evaluate the integrals for the volume of rotation using: a) the method of cylindrical shells; b) the method of circular disks.
stuck.

6. & 7. I'd just like to know if my answers are right.
 
  • #4
5a is wrong because the problem says 'bounded by the x-axis'. If the tangent line crosses the x-axis in the interval [0,1] then it's not the lower bound over the whole range of integration. Can we try and stick with one problem at a time?? Otherwise the confusion can be truly amazing.
 

What is the difference between the shell method and the disk method for calculating area and volume?

The shell method is used to calculate the volume of a solid of revolution by integrating the area of multiple cylindrical shells. The disk method, on the other hand, involves integrating the area of multiple disks to find the volume. The main difference between the two methods is the shape of the cross-section that is being integrated.

How do you determine the limits of integration for the shell and disk methods?

The limits of integration depend on the shape of the solid of revolution. For the shell method, the limits are typically determined by the boundaries of the region being rotated around the axis of revolution. For the disk method, the limits are determined by the points of intersection between the curve and the axis of revolution.

What is the formula for calculating the volume using the shell method?

The formula for the volume of a solid of revolution using the shell method is V = 2π∫(x)(f(x))dx, where x is the distance from the axis of revolution and f(x) is the function defining the curve being rotated. The integral is evaluated between the limits of integration.

What are the applications of the shell and disk methods in real life?

The shell and disk methods are commonly used in engineering and physics to calculate the volume and surface area of objects with rotational symmetry. They are also used in fields such as architecture, design, and manufacturing to determine the volume of curved or rotated objects.

How do you determine which method to use for a given solid of revolution?

The method used depends on the shape of the cross-section that is being integrated. If the cross-section is a disk, the disk method should be used. If the cross-section is a cylindrical shell, the shell method is more appropriate. In some cases, both methods can be used to double check the results.

Similar threads

Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
266
Find the volume using the shell method
    • Calculus and Beyond Homework Help
Replies
8
Views
1K
Verify Green's Theorem in the given problem
    • Calculus and Beyond Homework Help
Replies
10
Views
438
Volume of revolution, region bounded by two functions
    • Calculus and Beyond Homework Help
Replies
1
Views
912
Center of mass of spherical shell inside of cone (Apostol Problem).
    • Calculus and Beyond Homework Help
Replies
3
Views
557
Computing line integral using Stokes' theorem
    • Calculus and Beyond Homework Help
Replies
8
Views
874
Variation of parameter VS Undetermined Coefficients
    • Calculus and Beyond Homework Help
Replies
7
Views
497
Change of variables in multiple integrals
    • Calculus and Beyond Homework Help
Replies
34
Views
2K
Calculating the volume of the solid in this graph
    • Calculus and Beyond Homework Help
Replies
4
Views
944
Solid generated by revolving region, find the diameter of the hole
    • Calculus and Beyond Homework Help
Replies
1
Views
909

Hot Threads

Recent Insights

Back
Top