Mass of Surface: Find Double Integral over S

In summary, the problem asks for the double integral of the mass per unit area over the part of a cylinder in the first octant and within another cylinder. To solve this, one must visualize the shape and determine the appropriate bounds of integration. Looking at the shape from above and shading the region in the xy-plane can help with this process.
  • #1

Homework Statement


If the mass per unit area of a surface is given by rho = x*y, find the mass
double integral over S of x*y dS if S is the part of the cylinder x^2 + z^2 = 9 which is in the first octant and contained within the cylinder x^2 + y^2 = 4.

Homework Equations


Above.

The Attempt at a Solution


I can visualize what the shape looks like, but I'm not sure how to define the bounds of integration based on the shape described.
 
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  • #2
does anyone have any ideas? help would be greatly appreciated.
 
  • #3
Make sure you're visualizing the shape from a useful viewpoint. I suggest looking "from above." In other words, draw the xy-plane as usual on your paper, position your eye directly above the origin, and imagine the z-axis pointing straight out at your eye. Shade the region in the xy-plane that is covered by the surface.
 

What is the meaning of "Mass of Surface"?

The "Mass of Surface" refers to the total amount of matter present on a given surface. It takes into account both the surface area and the density of the material present on the surface.

What is a double integral?

A double integral is a type of mathematical operation used to calculate the volume under a surface in three-dimensional space. It involves integrating a function of two variables over a specific region on the surface.

Why is it important to find the mass of a surface?

Finding the mass of a surface is important in various scientific fields, such as physics, chemistry, and engineering. It allows us to understand the distribution of matter on a surface and can provide insights into the physical properties and behavior of the surface.

How do you calculate a double integral for finding the mass of a surface?

To calculate a double integral for finding the mass of a surface, you need to first define the region of the surface over which you want to find the mass. Then, you need to determine the density function, which represents the distribution of matter on the surface. Finally, you can use the double integral formula to calculate the mass of the surface by integrating the density function over the defined region.

What factors can affect the accuracy of the calculated mass of a surface?

The accuracy of the calculated mass of a surface can be affected by several factors, such as the accuracy of the measurement of the surface area and the density function, the complexity of the surface, and the precision of the integration method used. It is important to consider these factors and minimize errors to obtain a more accurate result.

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