Unit Circle Double Integral: Is 2π/3 the Answer?

I suspect, greater than 1 :smile:)In summary, the double integral discussed is \int\int_R sqrt(x^2+y^2) dx dy, where R is the unit circle. The integration is done as \int_0^\pi\int_{-1}^1 sqrt(r^2) r dr dtheta and the final answer is 2pi/3. However, it should be noted that the lower limit of the second integral should be -1 instead of 1.
  • #1
squenshl
479
4
For the double integral [tex]\int\int_R[/tex] sqrt(x^2+y^2) dx dy where R is the unit circle.
I got[tex]\int_0^\pi\int_1^1[/tex] sqrt(r2) r dr dtheta
Then after the integration I got an answer of 2[tex]pi[/tex]/3 as my final answer.
Is this right.
The bottom of the 2nd integral is -1 not 1
 
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  • #2
squenshl said:
For the double integral [tex]\int\int_R[/tex] sqrt(x^2+y^2) dx dy where R is the unit circle.
I got[tex]\int_0^\pi\int_1^1[/tex] sqrt(r2) r dr dtheta
Then after the integration I got an answer of 2[tex]pi[/tex]/3 as my final answer.
Is this right.
The bottom of the 2nd integral is -1 not 1


Hi squenshl! :smile:

(have a pi: π and a theta: θ and try using the X2 tag just above the Reply box :wink:)

(and you needed \int_{-1}^1)

erm :redface: … you can't have r less than 0! :wink:​
 

1. What is a unit circle double integral?

A unit circle double integral is a type of mathematical calculation that involves finding the area under a curve on a unit circle. It is a common concept in calculus and is often used to solve problems related to geometry, physics, and engineering.

2. What does the answer 2π/3 represent in a unit circle double integral?

The answer 2π/3 represents the area under a curve on a unit circle that is bounded by two angles, where one angle is π/3 and the other angle is 2π/3. It is the area of a sector of the unit circle.

3. How is the answer 2π/3 calculated in a unit circle double integral?

The answer 2π/3 is calculated by using the formula for the area of a sector of a circle, A = (θ/2) * r^2, where θ is the central angle and r is the radius of the circle. In this case, θ = 2π/3 and r = 1, so the calculation becomes A = (2π/3)/2 * 1^2 = 2π/3.

4. What is the significance of the answer 2π/3 in a unit circle double integral?

The answer 2π/3 represents the area of a sector of the unit circle, which has a central angle of 2π/3. It is important in calculus and other fields of mathematics as it can be used to solve various problems related to finding areas and volumes of curved shapes.

5. Is 2π/3 always the answer in a unit circle double integral?

No, 2π/3 is not always the answer in a unit circle double integral. The answer will vary depending on the specific problem and the values of the angles and radius involved. It is important to carefully evaluate the given problem and use the appropriate formulas to find the correct answer.

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