Question 1: Summarize the Riemann sum argument that gives the formula of area as a double integral.Hint 1: Read the subsection “Areas of Bounded Regions in the Plane” (p. 911 – 913). 15.3 Area by Double IntegrationIn this section we show how to use double integrals to calculate the areas of boundedregions in the plane, and to find the average value of a function of two variables.Areas of Bounded Regions in the PlaneIf we take f(x, y) = 1 in the definition of the double integral over a region R in the preced-ing section, the Riemann sums reduce toS₁₁ =FOX. 30 AA - AA.(1)This is simply the sum of the areas of the small rectangles in the partition of R, andapproximates what we would like to call the area of R. As the norm of a partition of Rapproaches zero, the height and width of all rectangles in the partition approach zero, andthe coverage of R becomes increasingly complete (Figure 15.8). We define the area of R tobe the limitο ΣΔΑlim||P0RdA.(2) 9120Chapter 15 Multiple Integrals(-1, 1),FIGURE 15.19 The region inExample 1.R₁0(1, 1)/y=x?(a)y=x+2,(2,4)for deR₂dx dydx dyChapter 15 Multiple IntegralsDEFINITION The area of a closed, bounded plane region R is- [[RAs with the other definitions in this chapter, the definition here applies to a greatervariety of regions than does the earlier single-variable definition of area, but it agrees withthe earlier definition on regions to which they both apply. To evaluate the integral in thedefinition of area, we integrate the constant function f(x, y) = 1 over R.A =EXAMPLE 1 Find the area of the region R bounded by y = x and y = x² in the firstquadrant.1- [SaydA.Solution We sketch the region (Figure 15.19), noting where the two curves intersect atthe origin and (1, 1), and calculate the area asEXAMPLE 2line y = x + 2.dy dx =- - L'[ - ], ₁²₁ - L' (xdx =(x - x²) dx =31-36Notice that the single-variable integral f(x-x²) dx, obtained from evaluating the insideiterated integral, is the integral for the area between these two curves using the method ofSection 5.6.Find the area of the region R enclosed by the parabola y = ² and theSolution If we divide R into the regions R, and R₂ shown in Figure 15.20a, we maycalculate the area as

The Riemann sum argument provides a method of integration for defining the area under a curve using…

Answered: Areas of Bounded Regions in the Plane

Math

Advanced Math

Areas of Bounded Regions in the Plane

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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