Chapter 15 – Multiple Integrals 15.4 Double Integrals in Polar Coordinates 1 Objectives: ...

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Transcript of Chapter 15 – Multiple Integrals 15.4 Double Integrals in Polar Coordinates 1 Objectives: ...

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Chapter 15 Multiple Integrals15.4 Double Integrals in Polar Coordinates15.4 Double Integrals in Polar Coordinates1Objectives:Determine how to express double integrals in polar coordinates

Dr. Erickson1Polar Coordinates ReviewRecall from this figure that the polar coordinates (r, ) of a point are related to the rectangular coordinates (x, y) by the equationsr2 = x2 + y2x = r cos y = r sin

15.4 Double Integrals in Polar Coordinates2

Dr. EricksonDouble Integrals in Polar CoordinatesSuppose that we want to evaluate a double integral , where R is one of the regions shown here.

15.4 Double Integrals in Polar Coordinates3

Dr. EricksonDouble Integrals in Polar CoordinatesIn either case, the description of R in terms of rectangular coordinates is rather complicated but R is easily described by polar coordinates.

15.4 Double Integrals in Polar Coordinates4

Dr. EricksonPolar RectangleThe regions in the first figure are special cases of a polar rectangle R = {(r, ) | a r b, } shown here.

15.4 Double Integrals in Polar Coordinates5

Dr. EricksonPolar RectangleTo compute the double integral

where R is a polar rectangle, we divide:

The interval [a, b] into m subintervals [ri1, ri] of equal width r = (b a)/m.

The interval [ ,] into n subintervals [ j1, j] of equal width = ( )/n.

15.4 Double Integrals in Polar Coordinates6

Dr. EricksonPolar SubrectangleThe center of the polar subrectangleRij = {(r, ) | ri1 r ri, j1 i}has polar coordinatesri* = (ri1 + ri)j* = (j1 + j)

15.4 Double Integrals in Polar Coordinates7

Dr. EricksonPolar RectanglesThe rectangular coordinates of the center of Rij are (ri* cos j*, ri* sin j*).

So, a typical Riemann sum is shown with Equation 1:

15.4 Double Integrals in Polar Coordinates8

Dr. EricksonPolar RectanglesIf we write g(r, ) = r f(r cos , r sin ), the Riemann sum in Equation 1 can be written as:

This is a Riemann sum for the double integral

15.4 Double Integrals in Polar Coordinates9

Dr. EricksonPolar RectanglesThus, we have:

15.4 Double Integrals in Polar Coordinates10

Dr. EricksonChange to Polar Coordinates15.4 Double Integrals in Polar Coordinates11

Dr. EricksonChange to Polar CoordinatesFormula 2 says that we convert from rectangular to polar coordinates in a double integral by:

Writing x = r cos and y = r sin Using the appropriate limits of integration for r and Replacing dA by dr d

Be careful not to forget the additional factor r on the right side of Formula 2.

15.4 Double Integrals in Polar Coordinates12Dr. EricksonChange to Polar CoordinatesA classical method for remembering the formula is shown here.

The infinitesimal polar rectangle can be thought of as an ordinary rectangle with dimensions r d and dr.

So, it has area dA = r dr d.

15.4 Double Integrals in Polar Coordinates13

Dr. EricksonExample 1Sketch the region whose area is given by the integral and evaluate the integral.15.4 Double Integrals in Polar Coordinates14

Dr. EricksonExample 2Evaluate the integral by changing to polar coordinates.15.4 Double Integrals in Polar Coordinates15

Dr. EricksonMore Complicated RegionsWhat we have done so far can be extended to the more complicated type of region shown here.

Its similar to the type II rectangular regions considered in Section 15.3

15.4 Double Integrals in Polar Coordinates16

Dr. EricksonMore Complicated VolumesIf f is continuous on a polar region of the form D = {(r, ) | , h1() r h2()}then

15.4 Double Integrals in Polar Coordinates17

Dr. EricksonVolumes and AreasIn particular, taking f(x, y) = 1, h1() = 0, and h2() = h() in the formula, we see that the area of the region D bounded by = , = , and r = h() is:

Which agrees with formula 3 in section 10.4.

15.4 Double Integrals in Polar Coordinates18

Dr. EricksonExample 3Use a double integral to find the area of the region.15.4 Double Integrals in Polar Coordinates19

Dr. EricksonExample 4 Use polar coordinates to find the volume of the given solid.15.4 Double Integrals in Polar Coordinates20

Dr. EricksonExample 5 Evaluate the iterated integral by converting to polar coordinates.15.4 Double Integrals in Polar Coordinates21

Dr. EricksonMore ExamplesThe video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. Example 2Example 3Example 415.4 Double Integrals in Polar Coordinates22Dr. EricksonDemonstrationsFeel free to explore these demonstrations below.Double Integral for Volume15.4 Double Integrals in Polar Coordinates23Dr. Erickson