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Double integrals in polar coordinates
- Intro Lesson: a5:11
- Intro Lesson: b11:54
- Lesson: 115:14
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- Lesson: 312:59
- Lesson: 426:15
Double integrals in polar coordinates
Lessons
Notes:
Review of Polar Coordinates
Double Integrals in Polar Coordinates
Review of Polar Coordinates
When converting from Cartesian Coordinates to Polar Coordinates, we say that:
x=rcosθ
y=rsinθ
We change all x's and y's into r's and θ's. We also use these formulas that could be useful for conversions:
x2+y2=r2
x2+y2=r
θ=tan−1xy
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 D:
∫∫Df(x,y)dA
Then we can convert it into polar coordinates such that:
∫∫Df(x,y)dA=∫θ=αθ=β∫r=g1(θ)r=g2(θ)f(rcosθ,rsinθ)rdrdθ
Why do we have to convert to polar coordinates? Watch the video and find out!
- IntroductionDouble Integrals in Polar Coordinates Overview:a)Review of Polar Coordinates
- change all x's and y's into r's and θ's
- x=rcosθ
- y=rsinθ
- An Example
b)Double Integrals in Polar Coordinates- Convert to r's and θ's
- Add an extra r
- Integrate in terms of r & θ
- An Example
- 1.Evaluating Double Integrals Using Polar Coordinates
Evaluate the double integral ∫∫D9x2+9y2dA where the region D is between the first quadrant of x2+y2=1 and x2+y2=4. - 2.Evaluate the double integral ∫∫Dx−ydA where the region D is the portion of x2+y2=4 in the second quadrant.
- 3.Evaluate the following double integral
∫−22∫04−x2ex2+y2dydx - 4.Use double integrals to determine the area of the region that is inside r=3+3sinθ and outside r=1−sinθ.