Double integrals in polar coordinates

Double integrals in polar coordinates

Lessons

Notes:

Review of Polar Coordinates

When converting from Cartesian Coordinates to Polar Coordinates, we say that:

x=rcosθ x = r \cos \theta
y=rsinθy = r \sin \theta

We change all xx's and yy's into rr's and θ\theta's. We also use these formulas that could be useful for conversions:

x2+y2=r2x^2 + y^2 = r^2
x2+y2=r\sqrt{x^2 + y^2} = r
θ=tan1yx \theta = \tan^{-1} \frac{y}{x}

Keep in mind that polar coordinates are useful when we come across circles or ellipses.


Double Integrals in Polar Coordinates

Suppose we have the following integral with region DD:

Df(x,y)dA\int \int_D f(x,y)dA

Then we can convert it into polar coordinates such that:

Df(x,y)dA=θ=αθ=βr=g1(θ)r=g2(θ)f(rcosθ,rsinθ)rdrdθ\int \int_D f(x,y)dA = \int^{\theta= \beta }_{\theta = \alpha} \int^{r=g_2(\theta )}_{r=g_1(\theta )} f(r \cos \theta, r \sin \theta ) rdrd\theta

Why do we have to convert to polar coordinates? Watch the video and find out!

  • Introduction
    Double Integrals in Polar Coordinates Overview:
    a)
    Review of Polar Coordinates
    • change all xx's and yy's into rr's and θ\theta's
    • x=rcosθ x = r \cos \theta
    • y=rsinθ y = r \sin \theta
    • An Example

    b)
    Double Integrals in Polar Coordinates
    • Convert to rr's and θ\theta's
    • Add an extra rr
    • Integrate in terms of rr & θ\theta
    • An Example


  • 1.
    Evaluating Double Integrals Using Polar Coordinates
    Evaluate the double integral D9x2+9y2dA\int \int_{D} \sqrt{9x^{2} + 9y^{2}} \, dA where the region DD is between the first quadrant of x2+y2=1 \, x^{2} + y^{2} = 1 \, and x2+y2=4 \, x^{2} + y^{2} = 4 .

  • 2.
    Evaluate the double integral DxydA\int \int_{D} x - y \, dA \, where the region DD is the portion of x2+y2=4 \, x^{2} + y^{2} = 4 \, in the second quadrant.

  • 3.
    Evaluate the following double integral

    2204x2ex2+y2dydx \large \int_{-2}^{2}\int_{0}^{\sqrt{4 - x^{2}}} \sqrt{e^{x^{2}+y^{2}}} \, dydx

  • 4.
    Use double integrals to determine the area of the region that is inside r=3+3sinθ \, r = 3 + 3 \, sin\theta\, and outside r=1sinθ \, r = 1 - \, sin\theta.