Calculators Definite Double Integral on Wolfram

Thread starter ahaanomegas
Start date Nov 22, 2011
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Double integral Integral

AI Thread Summary

To solve the problem of finding the double integral of (x^2 + y^2) over the circle defined by the equation x^2 + y^2 = 1, the approach involves using polar coordinates due to the symmetry of the circle. The integral can be transformed into polar coordinates where x = r cos(θ) and y = r sin(θ). The limits for r will be from 0 to 1, and for θ from 0 to 2π. The expression (x^2 + y^2) becomes r^2 in polar coordinates, leading to the integral being expressed as ∫ from 0 to 2π ∫ from 0 to 1 (r^2) * r dr dθ. This simplifies to evaluating the integral of r^3 with respect to r and then multiplying by the integral of dθ over the specified limits. The final result can be computed using Wolfram|Alpha or Mathematica for precise calculations.

Nov 22, 2011

ahaanomegas

How would you solve the following problem on Wolfram|Alpha (www.wolframalpha.com)?

Problem

If C is a circle or radius 1 with equation x^2+y^2=1, find

\iint_{C} (x^2 + y^2) \cdot \mathrm{d}x \cdot \mathrm{d}y

Last edited by a moderator: May 5, 2017

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Dec 19, 2011

Bill Simpson

Fall back on Mathematica notation

http://www.wolframalpha.com/input/?i=4*Integrate[x^2+++y^2,+{x,+0,+1},+{y,+0,+Sqrt[1+-+x^2]}]

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