Double Integrals in Polar Coordinates

Finding Area In Polar Coordinates
Finding Area In Polar Coordinates

Download presentation

Presentation is loading. Please wait.

1
Double Integrals in Polar Coordinates
15.4 Double Integrals in Polar Coordinates

2
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. 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 dr and 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
2. Charge Density and Charge
1. Density and Mass 2. Charge Density and Charge

16
2. Charge density and total Charge:
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 is C.

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 Using Formula 2 with f (x, y) = x2 + 2y, we get:
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:

Similar presentations

© 2023 SlidePlayer.com Inc.
All rights reserved.

You are watching: Double Integrals in Polar Coordinates. Info created by THVinhTuy selection and synthesis along with other related topics.

Rate this post

Related Posts