Download presentation

Presentation is loading. Please wait.

Published byCynthia Fowler Modified over 3 years ago

1
If f(x) 0 for a x b, then b f(x) dx a is defined to be the area of the region under the graph of f(x) on the interval [a,b]. The precise definition of this integral as the area is typically done by defining the integral as a limit of sums; this leads to the Fundamental Theorem of Calculus, which states that such integrals can be calculated using anti-differentiation. The definition of this integral can then be extended so that f(x) may possibly take on negative values (and areas below the x axis will be negative).

2
If f(x,y) 0 for all (x,y) in a region D of the xy plane, then the double integral is defined to be the volume under the graph of f(x,y) on the region D. When the region D is a rectangle R = [a,b] [c,d], then the volume is that of the following solid: (b,c) z (a,c) (a,d) (b,d) z = f(x,y) y x f(x,y) dx dy= f(x,y) dA DD

3
ExampleSuppose f(x,y) = k where k is a positive constant, and let R be the rectangle [a,b] [c,d]. Find (b,c) z (a,c) (a,d) (b,d) f(x,y) = k Since the integral is the volume of a rectangular box with dimensions b – a, d – c, and k, then the integral must be equal to y x k(b – a)(d – c). f(x,y) dA. R ExampleLet R be the rectangle [0,1] [0,1]. Find (1,1,0) x y z Since the integral is the volume of half of a unit cube, then the integral must be equal to f(x,y) = (1 – x) (1,0,0) (0,0,1)(0,1,1) (0,1,0) 1/2. (1 – x) dA. R

4
Cavalieri’s Principle : Suppose a solid is sliced by a series of planes parallel to the yz plane, each labeled P x, and the solid lies completely between P a and P b. z y x A(x)A(x) PxPx Let A(x) be the area of the slice cut by P x. If x represents the distance between any pair of parallel planes, then A(x) x is approximately the volume of Summing the approximations A(x) x is an approximation to the portion of the solid between P x and its successor. the volume of the solid. Taking the limit of the sum as x 0, we find that the solid’s volume is b A(x) dx a

5
Apply Cavalieri’s principle to find the volume of the following cone: length = 12 length = 13 circle of radius z x y 2.5 y = 5 — x 12 For 0 x 12, the slice of the cone at x parallel to the yz plane is a circle with diameter The area of the slice of the cone at x is A(x) = 5 — x. 12 25 —– x 2. 576 The volume of the cone is A(x) dx = a b 0 12 25 —– x 2 dx = 576 25

6
To apply Cavalieri’s principle to the double integral of f(x,y) over the rectangle R = [a,b] [c,d], we first find, for each value of x, the area A(x) of a slice of the solid formed by a plane parallel to the yz plane: (b,c) x y z (a,c) (a,d) (b,d) z = f(x,y) A(x) = Consequently, d f(x,y) dy c f(x,y) dA = R b A(x) dx = a b d f(x,y) dydx. ac By reversing the roles of x and y, we may write f(x,y) dA = R d b f(x,y) dxdy. ca

7
These equations turn out to be valid even when f takes on negative values. The integrals on the right side of each equation are called iterated integrals (and sometimes the brackets are deleted). ExampleLet R be the rectangle [–1,1] [0,1]. Find (x 2 + 4y) dA. R (x 2 + 4y) dA = R (x 2 + 4y) dy dx = 0 11 –1–1 x 2 y + 2y 2 dx = y = 0 1 1 –1–1 (x 2 + 2) dx = – 1 1 x 3 /3 + 2x = x = –1 1 14 — 3 Alternatively, one could find that (x 2 + 4y) dx dy = 0 11 –1–1 14 — 3

8
ExampleLet R be the rectangle [0, /2] [0, /2]. Find (sin x)(cos y) dA. R (sin x)(cos y) dA = R (sin x)(cos y) dy dx = 0 /2 0 (sin x)(sin y) dx = /2 0 y = 0 /2 (sin x) dx = 0 /2 – cos x = x = 0 /2 1 Alternatively, one could find that (sin x)(cos y) dx dy = 0 /2 0 1

Similar presentations

Presentation is loading. Please wait....

OK

Double Integrals over Rectangles

Double Integrals over Rectangles

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on advanced energy conversion systems Ppt on security features of atmosphere Ppt on working of human eye and defects of vision Ppt on 21st century skills curriculum Ppt on eia report oil Ppt on being creative youtube Ppt on real numbers for class 9th maths Ppt on general knowledge questions and answers Ppt on unemployment in india free download Ppt on range of hearing