Initially I would think you could just do a single integration along z because of the symmetry of the surfaces. Can you try making a couple sketches to help you understand the geometry? Do one sketch from the side (say, looking down the x-axis with the y-axis to the right and z upward), and do one sketch in a perspective view looking at an angle down from the side (like along the line x=y=z)...IAmGroot48 said:
Can you sketch a graph of the surface ##z = 4(x^2 + y^2)##? It's a paraboloid with its vertex at the origin, its axis along the z-axis, and opening upward. The two cylinders also have their axes along the z-axis.IAmGroot48 said:
Was this rhetorical?Mark44 said:
Parabolic!IAmGroot48 said:
Only that it's a parabolic question, rather than a rhetorical one.IAmGroot48 said:
Wow, nice sketch! What software package did you use to do that?IAmGroot48 said:
IAmGroot48 said:
Normally I use autocad. But this is a simple piece of software you can easily get for free. You can get it on Microsoft Store and it's called Graphing Calculator 3, there's a free version and a pro version.berkeman said:
Ill show you the work I've done so far. I think it finally clicked. "I think"PeroK said:
No, not at all. I didn't know you had access to a graphing tool.IAmGroot48 said:
I spoke with the Admin, and he will look at adding a Dark Mode in the 2022 updates.IAmGroot48 said:
My computers have an option to turn down the brightness. Have you tried that?Personally, I really dislike dark mode, as I find it harder to read, but hey, different strokes for different folks.IAmGroot48 said:
Or, in Latex: $$V = \int_0^{2\pi}\int_1^3\int_0^{4\rho^2}\rho \ dz \ d\rho \ d\phi = 160 \pi$$IAmGroot48 said:
Multi-integration in Calculus 3 refers to the process of calculating the volume of a solid or region in three-dimensional space using multiple integrals. It involves breaking down the three-dimensional shape into smaller, simpler shapes and integrating the function over each of these shapes to find the total volume.
The main difference between single and multi-integration in Calculus 3 is the number of variables involved. Single integration involves integrating a function with respect to one variable, while multi-integration involves integrating a function with respect to multiple variables. In Calculus 3, multi-integration is used to find the volume of three-dimensional shapes, while single integration is used for finding the area under a curve in two-dimensional space.
The volume of a solid or region can be calculated using multi-integration by breaking the three-dimensional shape into smaller, simpler shapes such as cylinders, spheres, or cones. The volume of each of these shapes can be calculated using single integration, and then the total volume can be found by summing up the volumes of all the smaller shapes.
Multi-integration in Calculus 3 has many real-world applications, such as in engineering, physics, and economics. It can be used to calculate the volume of complex three-dimensional objects, such as the volume of a water tank, the amount of material needed to construct a building, or the volume of a chemical solution in a container. It can also be used to find the center of mass of a three-dimensional object or to calculate the work done by a force on a three-dimensional object.
Some tips for solving multi-integration problems in Calculus 3 include breaking down the three-dimensional shape into smaller, simpler shapes, choosing the correct order of integration, and carefully setting up the limits of integration. It is also important to have a good understanding of single integration and the properties of integrals. Practicing with various examples and seeking help from a tutor or professor can also be helpful in mastering multi-integration problems.
