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15.4

15.4. Double Integrals in Polar Coordinates. Double Integrals in Polar Coordinates. Suppose that we want to evaluate a double integral:   R f ( x , y ) dA , where R is one of the regions shown below.

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15.4

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  1. 15.4 • Double Integrals in Polar Coordinates

  2. Double Integrals in Polar Coordinates • Suppose that we want to evaluate a double integral: Rf (x, y) dA, where R is one of the regions shown below. • In either case the description of R in terms of rectangular coordinates is rather complicated, but R is easily described using polar coordinates.

  3. Double Integrals in Polar Coordinates • Polar coordinates (r,  ) of a point are related to the rectangular coordinates (x, y) by the equations:

  4. Double Integrals in Polar Coordinates • Therefore we have: • Be careful not to forget the additional factor r on the right side! • A method for remembering this: • Think of the “infinitesimal” polar rectangle as an ordinary rectangle with dimensions • r d and drand therefore has “area” • dA = r dr d.

  5. Example 1 • Evaluate R (3x + 4y2) dA, where R is the region in the upper half-plane bounded by the circles x2 + y2 = 1 and x2 + y2 = 4. • Solution: • The region R can be described as: R = {(x, y) | y  0, 1 x2 + y2 4} • It is a half-ring and in polar coordinates it is given by: • 1 r 2, 0  .

  6. Example 1 – Solution • cont’d • Rewrite in terms of polar coordinates:

  7. Double Integrals in Polar Coordinates • What we have done so far can be extended to • the more complicated type of region • In fact, by combining Formula 2 with • where D is a type II region, we obtain the following formula: • Figure 7

  8. Practice 1: • Answer:

  9. Integrate in r: • Integrate in q:

  10. Practice 2: • Where D is the bottom half of : • Answer:

  11. Integrate in r: • Integrate in q:

  12. Practice 3: • Answer: • Region is defined by:

  13. So the integral is: • And we integrate in r then in q:

  14. Applications of Double Integrals • 15.5

  15. 1. Density and Mass • 2. Charge Density and Charge

  16. 1. Density and Mass: • Total mass m of an object with variable density (x, y): 2. Charge density and total Charge: • Total charge Q of a region D with charge density  (x, y) :

  17. Example • Charge is distributed over the triangular region D so that the charge density at (x, y) is  (x, y) = xy, measured in coulombs per square meter (C/m2). Find the total charge.

  18. Example – Solution • From Equation 2 and the graph of the region, we can write: The total charge isC.

  19. Another application of double integrals: • 15.6 • Surface Area

  20. Surface Area • Let S be a surface with equation z = f (x, y), where f has continuous partial derivatives:

  21. Surface Area • If we use the alternative notation for partial derivatives, we can rewrite Formula 2 as follows: • Notice the similarity between the surface area formula in Equation 3 and the arc length formula in calc 2:

  22. Example • Find the surface area of the part of the surface z = x2 + 2y that lies above the triangular region T in the xy-plane with vertices: (0, 0), (1, 0), and (1, 1). • Solution:The region T is shown below and is described by: • T = {(x, y) | 0 x  1, 0 y  x}

  23. Example – Solution • cont’d • Using Formula 2 with f (x, y) = x2 + 2y, we get: • Here is the graph of the area portion we have just computed:

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