Volume of F over R | Find Integral Bounds

• icesalmon
In summary, the homework statement says that the volume of the solid is bounded by the planes x=1 and x=2, the planes y= +/-(1/x) and below, and the planes z=x+1 and z=0. The attempt at a solution states that the best way to visualize the volume is to flat-out draw it in Mathematica. The double integral is then solved by integrating between two curves in x-y space, and the outer integral is solved by integrating between two points on the x-axis.

Homework Statement

Find the volume of the solid that is bounded on the front and back by the planes x = 1 and x = 2, on the sides by the cylinders y = +/-(1/x) and below by the planes z = x + 1 and z = 0

The Attempt at a Solution

I can't picture this thing at all, my sketch is so convoluted I am having trouble finding my bounds. If there's an easier way to get around this, I would love to see how. I do know that I have to use a double integral to compute V.

The best way is to just flat-out draw them in Mathematica. Otherwise, just need to practice drawing them. Make like only the picture is the homework and you are required to draw a nice one. Spend time doing just that and don't worry about the integral just yet. Then, easier to visualize this as a triple integral:

$$\int\int\int dzdydx$$

which reduces to a double since the integrand is just f(x,y,z)=1. So to integrate over z first, we integrate from some lower surface z=g(x,y), to some upper surface z=h(x,y). You said z goes from z=g(x,y)=0 to z=h(x,y)=x-1 so wouldn't we be integrating between those two surfaces? How then would we have to write the triple (or double) integral for that?

Now, for the double integral we next integrate between two curves in x-y space, from y=f1(x) to yj=f2(x). Well, again, you said y goes from -1/x to 1/x. Ok. Finally, for the outer integral, we integrate between two points on the x-axis and you gave that as well.

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okay, yeah it's been answered already, it's hard to picture a lot of these and obviously I'm not going to be able to use mathematica on my final. thanks!

1. What is the formula for finding the volume of F over R?

The formula for finding the volume of F over R is V = ∫F/R dx, where F is the function and R is the variable of integration.

2. How do you determine the bounds for the integral when finding the volume of F over R?

The bounds for the integral are determined by the limits of the variable of integration, which can be found by setting the function F equal to zero and solving for the variable R.

3. Can you explain the concept of volume of F over R in simpler terms?

The volume of F over R is a mathematical concept that represents the amount of space occupied by a three-dimensional object when rotated around an axis. It is calculated by finding the integral of the function F divided by the variable R.

4. What is the significance of finding the volume of F over R?

Finding the volume of F over R can be useful in various real-world applications, such as determining the volume of a container or calculating the amount of liquid or gas that can be held within a certain space.

5. Are there any special techniques or methods for finding the volume of F over R?

Yes, there are various techniques and methods that can be used to find the volume of F over R, such as the disk method, shell method, and cylindrical shells method. These methods involve breaking down the integral into smaller, more manageable pieces and using geometric principles to solve for the volume.