- #1
zhuyilun
- 27
- 0
Homework Statement
bounded by x^2+y^2=r^2 and y^2 +z^2=r^2
i guess r is just a random constant
Homework Equations
The Attempt at a Solution
i don't even have a clue of how to start this question
Use double integral to find the volume
- Thread starter zhuyilun
- Start date
In summary, the problem involves finding the volume bounded by two cylinders with radii r and equations x^2+y^2=r^2 and y^2+z^2=r^2. A helpful illustration has been provided and suggests using a yz integral. However, the axes may have been mislabeled and an xz integral may also be used.
bounded by x^2+y^2=r^2 and y^2 +z^2=r^2
i guess r is just a random constant
i don't even have a clue of how to start this question
- #2
- 9,568
- 775
I have posted a "freehand" picture to get you started. The picture shows the intersection of the cylinders in the first octant so it represents 1/8th of the volume. You will want to use a yz integral. I have oriented it so to make it "easier" to see.
[EDIT] Corrected mis-labeling of axes, use an xz integral
[EDIT] Corrected mis-labeling of axes, use an xz integral
Last edited:
- #3
PhaseShifter
- 279
- 1
Actually that illustration is for x2+y2=r2 and x2+z2=r2.
Not a huge deal though, since it gives the same numerical answer either way. Just rotate the axis labels one place in the clockwise direction and use an xz integral.
Not a huge deal though, since it gives the same numerical answer either way. Just rotate the axis labels one place in the clockwise direction and use an xz integral.
1. What is a double integral?
A double integral is a type of mathematical operation used to calculate the volume of a three-dimensional shape. It involves integrating a function over a two-dimensional region in order to find the volume under the surface of the function.
2. How is a double integral different from a single integral?
A single integral calculates the area under a curve in a two-dimensional plane, while a double integral calculates the volume under a surface in a three-dimensional space. This is because a single integral only has one variable, while a double integral has two variables.
3. When should I use a double integral to find volume?
A double integral should be used when the shape being measured has varying cross-sectional areas and cannot be easily broken down into simpler shapes. It is particularly useful for finding the volume of irregular shapes or objects with curved surfaces.
4. What are the steps to using a double integral to find volume?
The first step is to set up the double integral with the appropriate limits of integration, which define the boundaries of the region being integrated over. Next, the integrand, which represents the function being integrated, is multiplied by the infinitesimal area element to find the volume of each tiny slice. Finally, the entire integral is evaluated to find the total volume.
5. Are there any real-world applications of using a double integral to find volume?
Yes, double integrals are commonly used in physics, engineering, and other fields to find volumes of various shapes and objects. For example, they can be used to calculate the volume of a fluid in a container, the mass of a three-dimensional object, or the amount of material needed to fill a certain space.
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