Double Integral, where did I go wrong? Related to polar coordinates.

In summary, the correct answer for ∫∫cos(x^2 + y^2)dA, where R is the region that lies above the x-axis within the circle x^2 + y^2 = 9 is .5pi*sin(9). The mistake in the previous calculation was using a radius of 9 instead of 3.
  • #1
DavidAp
44
0
∫∫cos(x^2 + y^2)dA, where R is the region that lies above the x-axis within the circle x^2 + y^2 = 9.

Answer: .5pi*sin(9)

My Work:
∫(0 ->pi) ∫(0 -> 9) cos(r^2) rdrdθ

u = r^2
du = 2rdr
dr = du/2r

.5∫(0 ->pi) ∫(0 -> 9) cos(u) dudθ
.5∫(0 ->pi) sin(u)(0 -> 9) dθ
.5∫(0 ->pi) sin(r^2)(0 -> 9) dθ
.5∫(0 ->pi) [sin(9^2) - sin(0)] dθ
.5∫(0 ->pi) sin(81) dθ
.5[sin(81) θ](0 -> pi)
.5pi*sin(81)

However, the answer is suppose to be .5pi*sin(9). Where did I go wrong? Am I not suppose to square r, and if not then doesn't that mean everything I did below the substitution doesn't work?

Thank you for your time and patience. I apologize for my integral notation, I'm not sure of a better way to type it.
 
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  • #2
Your circle has radius 3, not 9. r should go from 0 to 3.
 
  • #3
O.O! I love you forever, thank you!
 

1. What are polar coordinates and how do they relate to double integrals?

Polar coordinates are a way of representing points in a two-dimensional coordinate system using a distance from the origin (r) and an angle (θ). Double integrals in polar coordinates are used to find the area of a region bounded by polar curves.

2. Why is it important to use polar coordinates in double integrals?

Polar coordinates can simplify the process of evaluating double integrals for certain types of regions, particularly circular or symmetric ones. They also allow for more efficient calculations in some cases.

3. What is the difference between a single integral and a double integral in polar coordinates?

A single integral in polar coordinates integrates over a curve, while a double integral integrates over a region in the polar plane. Additionally, a double integral in polar coordinates includes an extra factor of r (the distance from the origin) in the integrand.

4. How do I convert a double integral from rectangular to polar coordinates?

To convert a double integral from rectangular to polar coordinates, the limits of integration must be changed to reflect the new coordinate system. The integrand also needs to be multiplied by an extra factor of r. This factor is equivalent to the Jacobian determinant of the transformation.

5. What are some common mistakes to avoid when using polar coordinates in double integrals?

Some common mistakes include forgetting to include the extra factor of r in the integrand, using the wrong limits of integration, and not properly converting the integrand or the function being integrated. It is important to carefully follow the steps for converting and evaluating double integrals in polar coordinates to avoid these errors.

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