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motown
Lv 5
motown asked in Science & MathematicsMathematics · 1 decade ago

Multiple integrals help needed?

Need help with the following question:

Using polar coordinates, evaluate the following integral:

cos(x^2+y^2) dx dy over the region x^2+y^2< (or equal to) 9, x > (or equal to) 0.

I know that the region is the semi-circle, centre the origin with radius 3 above the x-axis and that after evaluating the jacobian dxdy becomes rdrd(theta) but how do you integrate rcos(r^2) dr.

Much appreciation goes to any answers.

1 Answer

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  • Aviram
    Lv 4
    1 decade ago
    Favourite answer

    ∫(θ -pi/2 to pi/2) ∫ (r 0 to 3) rcos(r^2) dr dθ=

    ∫(θ -pi/2 to pi/2) dθ * ∫ (r 0 to 3) rcos(r^2) dr=

    pi*∫ (r 0 to 3) rcos(r^2) dr=

    Substitute u=r^2, du=2rdr into the integral

    pi*∫ (u 0 to 9) 0.5*cos(u) du =

    pi/2 * sin(u) | (u - 0 to 9)=

    pi/2* sin(9)

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