Double Integrals In Polar Coordinates (HH: 16.4)

Transcription

1 Double Integrals In Polar Coordinates (HH: 16.4) In this lecture: 1. Motivating Double Integrals in Polar Coordinates 2. The Polar Area da = r dr dθ 3. Review of Polar Coordinates (Optional) Motivating Double Integrals in Polar Coordinates Much of the time the regions in our double integrals are circles, parts of circles or domains with some kind of circular symmetry. It turns out to be much easier to describe such regions using polar coordinates and double integrals are frequently easier to compute when this system is used. Let s look at such an integral in rectangular coordinates and then see how to tackle the same problem in polar coordinates. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 67

2 A circular plate 2 meters in radius and centered at the origin in the xy-plane. Suppose the mass-density at any point (x, y) is proportional to the distance of that point from the center of the plate, i.e., it gets heavier as you move toward the outer edge of the plate. Write a double integral (including all limits of integration) for the mass of the plate. Do not evaluate this integral. [Recall that M = δ(x, y) da] R Remark It would require large amounts of stamina for us to evaluate the integral that gives the mass of the plate in this problem, x 2 4 x 2 mass. k x 2 +y 2 dy dx. The answer is 16kπ/3 units of It might surprise you to know that this integral can be practically done in your head when using polar coordinates to describe the density function and region of integration. We will return to this later. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 68

3 Review of Polar Coordinates You are probably used to representing points in the plane using Cartesian coordinates, (x, y). However, points, P, in the plane can be represented in polar coordinates using the variables r and θ, where r is the distance of the point to the origin and θ is the angle with initial side along the positive x- axis and terminal side along the line segment OP. P r y O θ θ x Figure relating r, θ, x, and y The following equations relate polar and Cartesian coordinates: Conversion: Cartesian Polar r 2 =x 2 +y 2, tanθ=y/x; x 0 Conversion: Polar Cartesian x=r cosθ, y=r sinθ HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 69

4 Caution We caution you to take care in computing the angle θ. You will be using the arctangent function when you compute θ, but recall that the arctangent function only gives angles in the first and fourth quadrants. In the case that the point lies in the second or third quadrants, we must remember to add π to the arctangent angle to get the correct angle, θ. We demonstrate this situation in the next class activity. (A) Convert (x, y)=( 3, 4) into polar coordinates. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 70

5 (B) Convert (r, θ)=(4, π /6) into rectangular coordinates. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 71

6 (A) Sketch the graph of r=3. (B) Sketch the graph of θ=π /4. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 72

7 Sketch the graph of each equation and then convert each equation from rectangular form to polar form. (A) x 2 +y 2 =5 (B) x=2 (C) (x 0.5) 2 + y 2 = 1 4 HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 73

8 Remark When performing integrations involving circular regions or regions involving a circular sector, it is particularly useful to describe the regions using polar coordinates. (A) Sketch the following regions: (i) r 2, 0 θ π /2 (ii) 5 r 10, π / 3 θ 3π / 2 HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 74

9 (B) Describe each region using inequalities in polar coordinates. (i) 2 4 (ii) π / 6 π / 6 3 HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 75

10 Polar Rectangles and Polar Integrals We want to see what a double integral looks like when the region is in the shape of a polar rectangle shown. First we need to figure out how to find the area of such a rectangle. We explore this next. (A) Develop an expression for the area of a polar rectangle. [Hint: You can represent it as the difference of two polar sectors, one of inner radius r 1 the other of outer radius r 2.] HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 76

11 (B) Show that the area of the polar rectangle can be expressed as follows: A = r ΔrΔθ in which r is the average radius, r = r 1 +r 2 2. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 77

12 Remark Using the result of the previous activity, we develop the double integral in polar coordinates as follows. Suppose R is a polar rectangle defined by the inequalities a r b, α θ β, and g(r, θ) is a continuous function on R. We have partitioned the region into lots and lots of small pieces called polar rectangles (which are not really perfect rectangles at all). We assume that the rectangles are so small that r i r i+1 = r. Notice that each of the polar rectangles has an area given by ΔA r Δr Δθ 1 Definition If f is a continuous function on R, the polar rectangle a r b, α θ β and (r i, θ j ) is any point in the i, j-th polar rectangle, the definite integral of function g over region R is given by, g(r, θ) da = lim R r, θ 0 i, j g(r i, θ j )r r θ 1 Remember, θ must be in radians for this to work properly. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 78

13 Computation of an Integral in Polar Coordinates Just as with double integrals in rectangular coordinates, we can compute one in polar coordinates using an iterated integral as follows. Fubini's Theorem in Polar Coordinates Polar Rectangles Ifh(r, θ) is a continuous function on R, the polar rectangle α θ β, a r b, then, h(r, θ) da = h(r, θ) r dr dθ R β b α a Remark Note that this theorem applies to regions that are polar rectangles. Later we ll see a generalization that will work for more general polar regions in the plane. The additional r term comes from the fact that the area of the polar rectangle is r r θ. If you leave it out you will get the wrong answer! One way to remember this is to think of the equation da = dy dx for double integrals in rectangular coordinates, and the equation da = r dr dθ in polar coordinates. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 79

14 Rectangular to Polar Coordinate Conversion Formulas To convert a double integral in rectangular coordinates to an integral in polar coordinates, we will need the following coordinate conversion formulas: x = rcosθ, y = rsinθ, x 2 + y 2 = r 2 Use a double integral and Fubini's theorem in polar coordinates to evaluate, (x 2 +y 2 )da, where R is the region shown. R R 1 2 HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 80

15 Remark Note that the corresponding double integral in rectangular coordinates would look like this: 2 4 x 2 (x 2 +y 2 )dy dx (x 2 +y 2 )dy dx. Hideous! 2 4 x x 2 1 x 2 General Polar Regions Our sketch shows a fairly typical polar region in R 2. Assume that r = g 1 (θ) and r = g 2 (θ), then the region is described by the inequalities, α θ β, g 1 (θ) r g 2 (θ) Once again Fubini comes to our aid in the evaluation of these integrals. Here is Fubini's theorem for the double polar integral above. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 81

16 Fubini's Theorem for General Polar Regions Ifh(r, θ) is a continuous function on R, the polar region α θ β, g 1 (θ) r g 2 (θ), then, h(r, θ) da = h(r, θ) r dr dθ R β g (θ) 2 α g 1 (θ) We return to the problem we had at the start of this section. A circular plate 2 meters in radius and centered at the origin in the xy-plane. Suppose the mass-density at any point (x, y) is proportional to the distance of that point from the center of the plate. Find the mass of this plate. First set up the double integral in polar coordinates, and then evaluate it. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 82

17 Remark It's important to remember that double integrals in polar coordinates should be used only when they make an integral easier to compute. Similarly rectangular coordinates should be used if they prove more expedient. The next activity explores the idea of choosing the right coordinate system. For each region below, decide whether to use polar or rectangular coordinates for a double integral of the form f (x, y) da. Then set up an iterated integral with R appropriate limits of integration. 1 R 1 R HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 83

18 Sketch the region of integration for each double integral given in polar coordinates. (A) 2 1 π / 4 0 h(r, θ) r dθ dr (B) π / / cosθ 0 h(r, θ) r dr dθ HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 84

19 Evaluate the double integral e x2 +y 2 da where R is the unit disk 2 with center at the origin. R 2 The unit disk, is the set of all points satisfying x 2 + y 2 1. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 85

20 Now find the volume of the region bounded by the surface z = x 2 + y 2, the xy plane, and the cylinder x 2 + y 2 =16. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 86

21 Find the volume of the region bounded by the plane z =16 and surface z = x 2 + y 2. You will first need to make a sketch, find out what function f (x, y) you will be integrating, and then convert it to polar coordinates before actually performing the integration. HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 87

22 HH 16.4 Survival Guide Notes copyright 2014 Knobel/Stanley 88