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