Double integral over a circular region using rectangular coord's

In summary: This can also be thought of as the volume under the surface z = x^2 + y^2 over the region x^2 + y^2 ≤ 3.In summary, the conversation discusses the computation of a double integral using both polar and rectangular coordinates to find the area enclosed by a circle of radius 3 in the x-y plane. The calculations for both methods are provided and a mistake is discovered in the first attempt using polar coordinates. The significance of the double integral is also explained.
  • #1
sbpf
3
0
I would like to compute
$$ \iint \limits_{x^2 + y^2 \le 3} \! x^2 + y^2 \, \mathrm{d} A $$
using rectangular coord's.

First, I'll compute the iterated integral using polar coordinates so that I can check my work.

Limits:
$$ 0 \le \theta \le 2\pi \\
0 \le r \le 3 $$
so
$$ \iint \limits_{x^2 + y^2 \le 3} \! x^2 + y^2 \, \mathrm{d} A \
= \int_0^{2\pi} \!\!\! \int_0^3 \! ((r\cos\theta)^2 + (r\sin\theta)^2)r \, \mathrm{d}r\mathrm{d}\theta \
= \int_0^{2\pi} \!\!\! \int_0^3 \! r^3 \, \mathrm{d}r\mathrm{d}\theta \
= \frac{81\pi}{2} $$

Hopefully that's right.

Now for rectangular coord's:

I'll choose ##\mathrm{d}y\mathrm{d}x## as the order of integration.

Limits:
$$ -\sqrt{3} \le x \le \sqrt{3} \\
-\sqrt{3-x^2} \le y \le \sqrt{3-x^2} $$

so
$$ \iint \limits_{x^2 + y^2 \le 3} \! x^2 + y^2 \, \mathrm{d} A \
= \int_{-\sqrt{3}}^{\sqrt{3}} \! \int_{-\sqrt{3-x^2}}^{\sqrt{3-x^2}} \! x^2+y^2 \, \mathrm{d}y\mathrm{d}x \
= \frac{9\pi}{2} $$

so I've made a mistake somewhere...

To be honest, I'm not sure if I'm even close here. I'm guessing either my limits are wrong or I have absolutely no clue what I'm doing, so a point in the right direction would be greatly appreciated. Also, is there any significance to this particular double integral? We've just started covering multiple integration in class so I'm still trying grasp the concepts surrounding it.

Thanks.

Edit: I think I messed up my limits for the polar one previously by going to ##3## instead of ##\sqrt{3}##, could someone please verify that this newer one is correct?
$$ \iint \limits_{x^2 + y^2 \le 3} \! x^2 + y^2 \, \mathrm{d} A \\
= \int_0^{2\pi} \!\!\! \int_0^\sqrt{3} \! ((r\cos\theta)^2 + (r\sin\theta)^2)r \, \mathrm{d}r\mathrm{d}\theta \\
= \int_0^{2\pi} \!\!\! \int_0^\sqrt{3} \! r^3 \, \mathrm{d}r\mathrm{d}\theta \\
= \int_0^{2\pi} \! \frac{r^4}{4} \, \biggr|_{r=0}^\sqrt{3} \, \mathrm{d}\theta \\
= \int_0^{2\pi} \! \frac{9}{4} \, \mathrm{d}\theta \\
= \frac{9}{4}\theta \, \biggr|_{0}^{2\pi} \\
= \frac{9\pi}{2} $$
which is what I got for the rectangular coord's one, so I guess I was partially right?
 
Last edited:
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  • #2
To answer your question, the significance of this particular double integral is that it computes the area enclosed by a circle of radius 3 in the x-y plane.
 

1. What is a double integral over a circular region using rectangular coordinates?

A double integral over a circular region using rectangular coordinates is a mathematical calculation that involves finding the area under a surface within a circular region by dividing it into small rectangular sections and summing them up.

2. How is a double integral over a circular region using rectangular coordinates different from other types of integrals?

A double integral over a circular region using rectangular coordinates differs from other types of integrals because it involves integrating in two dimensions (x and y) instead of just one. It also requires a different approach and formula compared to single integrals.

3. What is the method for calculating a double integral over a circular region using rectangular coordinates?

The method for calculating a double integral over a circular region using rectangular coordinates involves dividing the circular region into small rectangles, finding the area of each rectangle, and then summing up all the areas to get an approximate value of the double integral. This process can be repeated with smaller and smaller rectangles to get a more accurate result.

4. What are some real-world applications of double integrals over a circular region using rectangular coordinates?

Double integrals over a circular region using rectangular coordinates have various real-world applications, such as calculating the mass or volume of objects with circular cross-sections, finding the center of mass of circular objects, and determining the electric potential or gravitational potential of circular distributions.

5. What are some common mistakes to avoid when solving a double integral over a circular region using rectangular coordinates?

Some common mistakes to avoid when solving a double integral over a circular region using rectangular coordinates include forgetting to properly set up the integral, not taking into account the correct limits of integration, and incorrectly calculating the area of the rectangular sections. It is also important to pay attention to the order of integration and to double-check calculations for accuracy.

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