Calculus 3 Sections 15.1 - 15.2 Reading Assignment: Double Integrals

Question 2: Explain each step of the procedure for finding the limits of integration when integrating in the order dx dy.

Hint 1: Read the subsection "Finding Limits of Integration" (p. 906).

Hint 2: Read Theorem 2 (boxed on p. 904) for a technical version for this question.

906 Chapter 15 Multiple Integrals Although Fubini's Theorem assures us that a double integral may be calculated as an iterated integral in either order of integration, the value of one integral may be easier to 1 find than the value of the other. The next example shows how this can happen. EXAMPLE 2 Calculate sin.N dA, FIGURE 15.13 The region of integration in Example 2. where R is the triangle in the xy-plane bounded by the x-axis, the line y = x, and the line r= 1. Solution The region of integration is shown in Figure 15.13. If we integrate first with respect to y and next with respect to x, then because x is held fixed in the first integration, we find I' ( ['sinzay ) as - / y sing a = sin xdx = -cos(1) + 1 = 0.46. rty=l If we reverse the order of integration and attempt to calculate R sin & dx dy. we run into a problem because / ((sinx)/x) dx cannot be expressed in terms of elemen- tary functions (there is no simple antiderivative) There is no general rule for predicting which order of integration will be the good one in circumstances like these. If the order you first choose doesn't work, try the other. Some- times neither order will work, and then we may need to use numerical approximations. " Leaves at Enters at Finding Limits of Integration We now give a procedure for finding limits of integration that applies for many regions in the plane. Regions that are more complicated, and for which this procedure fails, can often be split up into pieces on which the procedure works.Using Vertical Cross-Sections When faced with evaluating /, f(x, y) dA, integrat- (b) ing first with respect to y and then with respect to x, do the following three steps: 1. Sketch. Sketch the region of integration and label the bounding curves (Figure 15.14a). Leaves at 2. Find the y-limits of integration. Imagine a vertical line L cutting through R in the direc- tion of increasing y. Mark the y-values where & enters and leaves. These are the y-limits of integration and are usually functions of x (instead of constants) (Figure 15.14b). Enters al 3. Find the x-limits of integration. Choose x-limits that include all the vertical lines through R. The integral shown here (see Figure 15.14c) is Smallest x Largest I is x = 1 Using Horizontal Cross-Sections To evaluate the same double integral as an iterated (C) integral with the order of integration reversed, use horizontal lines instead of vertical lines in Steps 2 and 3 (see Figure 15.15). The integral is FIGURE 15.14 Finding the limits of integration when integrating first with respect to y and then with respect to x. ((x. y) de dy.904 Chapter 15 Multiple Integrals r = a, x = b, we may again calculate the volume by the method of slicing. We first calculate the cross-sectional area A(x) = f(x, y) dy and then integrate A(x) from x = a to x = b to get the volume as an iterated integral: why' V f(x. y) dy dx. (1) FIGURE 15.11 The volume of the solid Similarly, if R is a region like the one shown in Figure 15.11, bounded by the curves shown here is x = hy(y) and x = h (y) and the lines y = c and y = d, then the volume calculated by slicing is given by the iterated integral For a given solid, Theorem 2 says we can Volume = (2) calculate the volume as in Figure 15.10, or in the way shown here. Both calculations That the iterated integrals in Equations (1) and (2) both give the volume that we have the same result. defined to be the double integral of f over A is a consequence of the following stronger form of Fubini's Theorem.THEOREM 2-Fubini's Theorem (Stronger Form) Let f(x, y) be continuous on a region R. 1. If R is defined by a = x = b, g (x) =y = 2 (x), with g and g, continuous on [ a, b ] , then fix, y) dA = fox, y) dy dr. 2. If R is defined by c s y = d, h(y) = x = by(y), with h, and h, continuous on [ c, d ] , then EXAMPLE 1 Find the volume of the prism whose base is the triangle in the xy-plane bounded by the x-axis and the lines y = x and x = 1 and whose top lies in the plane z = [(x y) = 3 - x - y. Solution See Figure 15.12. For any x between ( and 1, y may vary from y = 0toy = x (Figure 15.12b). Hence, v= / 3 -x - maydx = ][3 - x-2ax