How can I improve my limit-setting skills for double integrals?

[mkkrnfoo85]

I took a test today. I wanted to know if I set my limits up correctly and got the right answer, because I've been having problems with that. Okay, here is the question:

A space is bounded by x = 0, y = 0, xy-plane, and the plane: 3x + 2y + z = 6. Find the volume using a double integral.

So, this is how I went about the problem...

Since the space is bounded by the xy-plane, I set z = 0 for the plane. This gave me:

$$3x + 2y = 6, y = \frac{-3x+6}{2}$$

That's just an equation of a line, so I plotted that on the xy-plane.

At x = 0, y = 3 (0,3)
At y = 0, x = 2 (2,0)

This is the hard part for me...setting limits. I got:

$$0\leq x\leq 2$$

and

$$\frac{-3x+6}{2}\leq y\leq 0$$
(hopefully)

*sidenote: If it's wrong, would someone like to show me a simple strategy to setting limits? Also, if there's anything else you could do to help me set limits on integration, that would be really helpful.

To go on with the problem, my resulting double integral was:

$$\int_{x=0}^{x=2} \int_{y= \frac{-3x+6}{2}}^{y=0} (-3x-2y+6)dydx$$

integrating with respect to y first, I got:

$$\int_{x=0}^{x=2} (-\frac{9}{4} x^2 + 9x-18)dx$$

resulting in answer = 9

Thanks for reviewing.

-Mark

(yay, I learned how to LaTeX :) )

 

[mkkrnfoo85]

oops, i think this belongs in the homework section... :(.

 

[M98Ranger]

First of all, congratulations on learning how to use LaTeX! It's a great tool for writing math equations and expressions.

Now, let's take a look at your solution to the problem. It seems like you have set up the limits correctly for the double integral. You have correctly identified the bounds for x and y, and your integration looks correct as well. So, good job on that!

As for setting limits in general, the best strategy is to always start by looking at the given boundaries and visualizing them on a graph. This will help you understand the shape of the region you are trying to find the volume of. Then, try to break down the region into smaller, simpler shapes (like rectangles or triangles) and set up the limits accordingly. It might take some practice, but with time and experience, you'll become more comfortable with setting limits for integration.

One tip that might help is to think about the order in which you integrate. In this case, you have chosen to integrate with respect to y first and then x. This is a good choice because the region is easier to visualize and set up in terms of y. However, if you had chosen to integrate with respect to x first, you would have to split the region into two parts (above and below the line y = -3x + 6) and set up separate limits for each part. So, it's always a good idea to think about which variable will be easier to integrate with respect to and choose the order accordingly.

Overall, your solution looks good and it seems like you have a good understanding of the Double Integral Test. Keep practicing and you'll become more confident in setting limits and solving integration problems. Good luck!