16.2 Iterated Integrals

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1 6.2 Iterated Integrals

2 So far: We have defined what we mean by a double integral. We have estimated the value of a double integral from contour diagrams and from tables of values. We have interpreted the computation of a double integral as volume. We have related the double integral to the concept of Average Value of a function.

3 Now we study how we can compute exactly the value of a double integral.

4 Example. Estimate total fox population from the following contour diagram of the population density. How to estimate total fox population? Population density of foxes in southwestern England.

5 Expressing a double integral as an iterated integral: Estimates of the fox population density: We computed: Total Population f ( u, v ) x y Now we try adding across the rows first: Total Population f ( uij, vij ) x y j i i, j ij ij

6 Total Population f ( uij, vij ) x y j i We observe that the inner sum approximates f ( x, v ) dx 8 ij ( 8 ) ij Thus, Total Population f ( x, v ) dx y j The outer sum approximates an integral whose integrand is 8 f ( x, v ) dx. ij Thus, we can find the total population in terms of iterated one-variable integrals: ( ) 5 8 Total Population = f ( x, y) dx dy

7 Theorem: If R us the rectangle a x b, c y d and f is a continuous functionon R, then the integral of f over R exists and is equal to the iterated integral: R fda y= d = y= c ( x= b ) (, ) x= a f x y dx dy Note: To evaluate the integral, first perform the inside integral with respect to x, holding y constant; then integrate with respect to y.

8 Example : A building is 8 meters wide and 6 meters long. It has a flat roof that is 2 meters high at one corner, and meters high at each of the adjacent corners. What is the volume of the building? Start by finding the equation of the roof.

9 The equation of the roof is z = 2 x y 4 8 The volume is given by Volume = 2 x y da 4 8 R What is the region R? Describe it using inequalities in x and y.

10 R is the rectangle x 8, y 6 Thus the volume is given by the iterated integrals: Volume 6 8 = 2 x 4 8 y dxdy Solve the inner integral (remember to consider y as a constant)

11 2 6 y Volume = ( 88 y) dy = 88y = What about if we try to integrate first with respect to y and then with respect to x? Example 2. Compute the volume with the following iterated integrals: Volume 8 6 = 2 x 4 8 y dydx

12 For any integral we will find in this course we have that the order of integration does not matter: R ( ) ( ) (, ) (, ) y= d x= b x= b y= d fda = f x y dx dy = f x y dy dx y= c x= a x= a y= c

13 Example 3. The density at the point (x,y) of a triangular metal plate, as shown in figure, is δ( x, y). Express its mass as an iterated integral Solution: Dividing the triangular region in small rectangles, we see that in each rectangle: Mass of rectangle δ ( x, y) x y

14 ( y d ) δ y c x= b = Total mass of triangle = ( x, y) dy dx x= a = We observe that the upper limit of y is 2-2x ( ) δ y x= y= 2 2x Total mass of triangle = ( x, y ) dy dx x= = Express the total mass by first integrating with respect to x

15 ( ) δ x y= 2 x= /2 y Total mass of triangle = ( x, y ) dx dy y= =

16 Example 4. Find the mass M of a metal plate R bounded by y = x y = x δ x y = + xy 2 2 and, with density given by (, ) kg/meter. M = R δ ( x, y) da * We integrate along vertical strips first. 2 * y lower boundary is y = x while its upper boundary is y = x. * On the other hand, x goes from to. Write the expression for the iterated integrals.

17 M x x = + xy dydx 2 ( ) Solve the integral!

18 Concept Question : The integral x 2 dxdy represents the (a) Area under the curve y = x 2 between x= and x=. (b) Volume under the surface z = x 2 above the square x, y on the xy-plane. (c) Area under the curve y = x 2 above the square x, y on the xy-plane.

19 ANSWER (b). The volume under the surface z = x 2 above the square x, y on the xy-plane.

20 Concept Question 2: The integral x dydx represents the (a) Area of a triangular region in the xy-plane. (b) Volume under the plane z = above a triangular region of the plane. (c) Area of a square in the xy-plane.

21 ANSWER (a) and (b) are both right.

22 Let f(x, y) be a positive function. Rank the following integrals from smallest to largest.. ), ( ) (, ), ( ) (, ), ( ) ( 3 2 dydx y x f c dydx y x f b dydx y x f a x x Concept Question 3:

23 ANSWER (a) is smaller than (b) which is smaller than (c). Since the regions are nested and the integrands are the same and positive, the only thing that matters is the size of the region we integrate over.

24 Match the integral with the appropriate region of integration. (a) R : The triangle with vertices (, ), (2, ), (, ). (b) R 2 : The triangle with vertices (, ), (, 2), (, ). (c) R 3 : The triangle with vertices (, ), (2, ), (2, ). (d) R 4 : The triangle with vertices (, ), (, ), (, 2). (i) (ii) (iii) (iv) 2 2x 2 2 y 2x 2 2 y Concept Question 4: f ( x, y) dydx f ( x, y) dxdy f ( x, y) dydx f ( x, y) dxdy

25 ANSWER (a) and (ii); (b) and (i); (c) and (iv); (d) and (iii)

26 Concept Question 5: Which of the following integrals is (are) equal to 3 4x f ( x, y) dydx? (a) 4x 3 f ( x, y) dxdy (b) 2 3 y/4 f ( x, y) dxdy (c) (e) 2 4x y/ f f ( x, ( x, y) dxdy y) dydx (d) 2 y/4 f ( x, y) dxdy

27 ANSWER (b) 2 3 y/4 f ( x, y) dxdy

Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.

Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral