1. Da Nang-09/2013 Natural Science Department – Duy Tan University Lecturer: Ho Xuan Binh Double Integrals in Polar Coordinates. We will learn about: How to express double integrals in polar coordinates.

2. Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University Suppose that we want to evaluate a double integral:, where R is one of the regions shown here.

3. Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University In either case, the description of R in terms of rectangular coordinates is rather complicated but R is easily described by polar coordinates.

4. Recall from this figure that the polar coordinates (r, θ) of a point are related to the rectangular coordinates (x, y) by the equations: r 2 = x 2 +y 2 x = r cos θ y = r sin θ Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University Polar coordinates *

5. Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University The regions in the first figure are special cases of a polar rectangle R = {(r, θ) | a ≤ r ≤ b, α ≤ θ ≤ β} shown here. Polar rectangle *

6. Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University If f is continuous on a polar rectangle R given by 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β where 0 ≤ β – α ≤ 2π, then Formula *

7. Change to polar coords * Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University We convert from rectangular to polar coordinates in a double integral by: *Replacing x by rcosθ and y by rsinθ *Replacing dA by rdrdθ

8. Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University Example * where R is the region in the upper half-plane bounded by the circles x 2 + y 2 = 1 and x 2 + y 2 = 4. Evaluate

9. Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University Example *

10. Double Integrals in Polar Coordinates Natural Science Department – Duy Tan University Example *

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