Math Double Integrals in Polar Coordinates

Transcription

1 Math Double Integrals in Polar Coordinates Peter A. Perry University of Kentucky October 22, 2018

2 Homework Re-read section 15.3 Begin work on 1-4, 5-31 (odd), 35, 37 from 15.3 Read section 15.4 for Wednesday, October 24 Continue working on Webwork C1

3 Unit III: Multiple Integrals Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture 29 Lecture 30 Lecture 31 Lecture 32 Lecture 33 Lecture 34 Double Integrals over Rectangles Double Integrals over General Regions Double Integrals in Polar Coodinates Applications of Double Integrals Surface Area Triple Integrals Triple Integrals in Cylindrical Coordinates Triple Integrals in Spherical Coordinates Change of Variable in Multiple Integrals, Part I Change of Variable in Multiple Integrals, Part II Exam III Review

4 Goals of the Day Review Polar Coordinates, introduce Polar Rectangles Learn how to compute double integrals over polar rectangles Learn how to compute double integrals over polar regions Learn to compute volumes using polar integrals

5 Reality Check Calculus I Calculus III Riemann sum Riemann Integral n f (xi ) x i=1 b a f (x) dx n f (xi, y j ) A i,j=1 f (x, y) da D Way of computing F (b) F (a) Iterated Integral Interpretation Area under a curve Volume under a surface

6 y Review of Polar Coordinates x 2 + y 2 = 1 r y x θ x y Recall that r 2 = x 2 + y 2, tan θ = y x x 2 + y 2 = 4 x and x = r cos θ, y = r sin θ. x 2 + y 2 = 1 How would you describe the regions at left in polar coordinates?

7 Polar Rectangles A polar rectangle is a region R = {(r, θ) : a r b, α θ β}.

8 Polar Rectangles A polar rectangle is a region R = {(r, θ) : a r b, α θ β}. r = b r = a

9 Polar Rectangles A polar rectangle is a region R = {(r, θ) : a r b, α θ β}. θ = β r = b r = a θ = α

10 Polar Rectangles A polar rectangle is a region R = {(r, θ) : a r b, α θ β}. Like an ordinary rectangle a polar rectangle can be divided into subrectangles

11 Polar Rectangles A polar rectangle is a region R = {(r, θ) : a r b, α θ β}. Like an ordinary rectangle a polar rectangle can be divided into subrectangles A small polar rectangle has area A r r θ r r θ

12 Integrals Over Polar Rectangles The double integral f (x, y) da is a R limit of Riemann sums: n f (ri cos θj, r i sin θj )r j r θ i,j=1 θ = β r = b (r i, θ j ) Rectangle R ij is given by R ij = {(r, θ) : r i 1 r r i, θ j 1 θ θ j } r = a θ = α where r i = a + i r, θ j = α + j θ r = b a n, θ = β α n In the limit this leads to an iterated integral β b f (r cos θ, r sin θ) r dr dθ α a

13 Integrals Over Polar Rectangles Double Integral In Polar Coordinates The integral of a continuous function f (x, y) over a polar rectange R given by a r b, α r β, is R f (x, y) da = β b f (r cos θ, r sin θ) r dr dθ α a 1. Find (2x y) da if R is the region in the first quadrant bounded by R the circle x 2 + y 2 = 4 and the lines x = 0 and y = x. 2. Find e x 2 y 2 da if D is the region bounded by the semicircle R x = 4 y 2 and the y-axis.

14 Integrals over Polar Regions θ = β h 1 (θ) h 2 (θ) θ = α If f is continuous over a polar region of the form D = {(r, θ) : α θ β, h 1 (θ) r h 2 (θ)} then f (x, y) da = D β h2 (θ) f (r cos θ, r sin θ) r dr dθ α h 1 (θ)

15 Integrals over Polar Regions θ = β h 1 (θ) h 2 (θ) y θ = α If f is continuous over a polar region of the form D = {(r, θ) : α θ β, h 1 (θ) r h 2 (θ)} then f (x, y) da = D β h2 (θ) f (r cos θ, r sin θ) r dr dθ α h 1 (θ) x Find the area of one loop of the rose r = cos 3θ

16 Volumes of Solids Find the volume under the paraboloid and above the disc z = x 2 + y 2 50 x 2 + y 2 < Describe the disc in polar coordinates 2. Transform f (x, y) to polar coordinates

17 Volumes of Solids Find the volume inside the sphere x 2 + y z + z 2 = 16 2 and outside the cylinder x 2 + y 2 = 4