# Volume Integration: Find Solutions

• blumfeld0
In summary, the conversation was about setting up integrals for finding the volume of solids obtained by rotating regions about specified lines. The first problem involved rotating the region between two circles, with radii 3 and y+3, about the line x=-3. The correct integral is pi times the integral from 3 to 4 of [(\frac{1}{x^3}+3)^2- 9]dx. The second problem involved rotating the region between two parabolas about the line x=-3. The correct integral is also pi times the integral from 0 to 1 of [(3-y^2)^2 - (3-\sqrt{y})^2]dy.
blumfeld0
hi guys. i just need help setting up the integral for the following two problems.

1. find the volume of the solid obtained by rotating the region by the given area about the specified lines.
y= 1/ x^3, y= 0, x=3, x=4 ABOUT x=-3

so i have integral from 3 to 4 of [(-3)^2 - (1/x^3)^2] dx

is that right?

2. find the volume of the solid obtained by rotating the region bounded by the given curves about specified line
y=x^2, x=y^2 about the line x=-3

so i have integral from 0 to 1 of [(3-x^2)^2 - (3-Sqrt(x))^2] dx

is that even close to right?

thank you

blumfeld0 said:
hi guys. i just need help setting up the integral for the following two problems.

1. find the volume of the solid obtained by rotating the region by the given area about the specified lines.
y= 1/ x^3, y= 0, x=3, x=4 ABOUT x=-3

so i have integral from 3 to 4 of [(-3)^2 - (1/x^3)^2] dx

is that right?
No, but it's close. For each x, you are looking at the region between two circles. The inner circle has radius 3 and the outer circle has radius y+3. Also you have forgotten the $\pi$ in $\pi r^2$! Your integral should be
$$\pi \int_3^4 [(\frac{1}{x^3}+3)^2- 9]dx$$

2. find the volume of the solid obtained by rotating the region bounded by the given curves about specified line
y=x^2, x=y^2 about the line x=-3

so i have integral from 0 to 1 of [(3-x^2)^2 - (3-Sqrt(x))^2] dx
Although, because of the symmetry, that will give you the correct answer, since you are rotating around the vertical line x= -3, your "washers" should be going vertically, so every "x" should be "y". Oh, and you forgot $\pi$ again!

is that even close to right?

thank you

For the first problem, your set up for the integral is correct. However, there are a few errors in the expression inside the integral. It should be (-1/x^3)^2 instead of (1/x^3)^2 and the limits of integration should be -3 to -4 since the region is being rotated around the line x=-3. The final integral should be:

V = ∫ from -3 to -4 of [(3^2) - (-1/x^3)^2] dx

For the second problem, your set up is also correct. However, the expression inside the integral should be (x^2)^2 instead of (3-x^2)^2 and (y^2)^2 instead of (3-Sqrt(x))^2. Also, the limits of integration should be from 0 to 1 since the region is being rotated around the line x=-3. The final integral should be:

V = ∫ from 0 to 1 of [(x^2)^2 - (y^2)^2] dy

Overall, your approach to setting up the integrals is correct, but make sure to double check the expressions and limits of integration to ensure the correct solution.

## 1. What is volume integration?

Volume integration is a mathematical method used to find the volume of a three-dimensional object by breaking it down into infinitesimally small pieces and summing the volumes of those pieces. It is an important tool in physics, engineering, and other fields where volume calculations are needed.

## 2. How is volume integration used in real life?

Volume integration is used in various real-life applications, such as calculating the volume of a liquid in a container, determining the amount of material needed for construction projects, and analyzing the structure of biological tissues and organs.

## 3. What are the steps involved in volume integration?

The first step is to set up a three-dimensional coordinate system and choose an appropriate shape to divide the object into smaller parts, such as cubes or cylinders. Then, the volume of each small piece is calculated using the appropriate formula. Finally, all the volumes are added together to get the total volume of the object.

## 4. What are the key concepts in volume integration?

The main concepts in volume integration include the use of triple integrals, the concept of infinitesimal volumes, and the application of the fundamental theorem of calculus. Additionally, understanding the geometry of the object and its boundaries is crucial in setting up the integral equations.

## 5. Are there any limitations to volume integration?

Volume integration has some limitations, such as the need for a well-defined shape and boundaries for the object, the complexity of the equations involved, and the need for advanced mathematical knowledge to perform the calculations. It is also not suitable for objects with irregular shapes or varying densities.

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