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  • Double Integrals in Polar Coordinates
    https://math.libretexts.org/Courses/Georgia_State_University_-_Perimeter_College/MATH_2215%3A_Calculus_III/15%3A_Multiple_Integration/Double_Integrals_in_Polar_Coordinates
    \[\begin{align*} \iint_D r^2 \sin \, \theta \, r \, dr \, d\theta &= \int_{\theta=0}^{\theta=\pi} \int_{r=0}^{r=1+\cos \theta} (r^2 \sin \, \theta) \,r \, dr \, d\theta \\ &= \frac{1}{4}\left.\int_{\t...∬Dr2sinθrdrdθ=∫θ=πθ=0∫r=1+cosθr=0(r2sinθ)rdrdθ=14∫θ=πθ=0[r4]|r=1+cosθr=0sinθdθ=14∫θ=πθ=0(1+cosθ)4sinθdθ=−14[(1+cosθ)55]π0=85.
  • 4.8: Double Integrals in Polar Coordinates
    https://math.libretexts.org/Courses/Misericordia_University/MTH_226%3A_Calculus_III/Chapter_15%3A_Multiple_Integration/4.08%3A_Double_Integrals_in_Polar_Coordinates
    \[\begin{align*} \iint_D r^2 \sin \, \theta \, r \, dr \, d\theta &= \int_{\theta=0}^{\theta=\pi} \int_{r=0}^{r=1+\cos \theta} (r^2 \sin \, \theta) \,r \, dr \, d\theta \\ &= \frac{1}{4}\left.\int_{\t...∬Dr2sinθrdrdθ=∫θ=πθ=0∫r=1+cosθr=0(r2sinθ)rdrdθ=14∫θ=πθ=0[r4]|r=1+cosθr=0sinθdθ=14∫θ=πθ=0(1+cosθ)4sinθdθ=−14[(1+cosθ)55]π0=85.
  • Double Integrals in Polar Coordinates
    https://math.libretexts.org/Courses/Montana_State_University/M273%3A_Multivariable_Calculus/15%3A_Multiple_Integration/Double_Integration_with_Polar_Coordinates/Double_Integrals_in_Polar_Coordinates
    \[\begin{align*} \iint_D r^2 \sin \, \theta \, r \, dr \, d\theta &= \int_{\theta=0}^{\theta=\pi} \int_{r=0}^{r=1+\cos \theta} (r^2 \sin \, \theta) \,r \, dr \, d\theta \\ &= \frac{1}{4}\left.\int_{\t...∬Dr2sinθrdrdθ=∫θ=πθ=0∫r=1+cosθr=0(r2sinθ)rdrdθ=14∫θ=πθ=0[r4]|r=1+cosθr=0sinθdθ=14∫θ=πθ=0(1+cosθ)4sinθdθ=−14[(1+cosθ)55]π0=85.

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