Michael Corral: Vector Calculus

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104 CHAPTER 3. MULTIPLE INTEGRALS area above the curve but below the x-axis when f(x)≤0. Similarly, the double integral of any continuous function f(x,y) represents the difference of the volume below the surface z= f(x,y) but above the xy-plane when f(x,y)≥0, and the volume above the surface but below the xy-plane when f(x,y)≤0. Thus, our method of double integration by means of iterated integrals can be used to evaluate the double integral of any continuous function over a rectangle, regardless of whether f(x,y)≥0or not. Example 3.3. Evaluate ∫ 2π ∫ π 0 0 sin(x+y)dxdy. Solution: Note that f(x,y)=sin(x+y) is both positive and negative over the rectangle [0,π]×[0,2π]. We can still evaluate the double integral: ∫ 2π ∫ π 0 0 sin(x+y)dxdy= ∫ 2π 0 ∫ 2π ( −cos(x+y) ∣ ∣ x=π x=0 ) dy = (−cos(y+π)+cosy)dy 0 =−sin(y+π)+siny ∣ 2π =−sin3π+sin2π−(−sinπ+sin0) 0 = 0 ☛ ✟ ✡Exercises ✠ A For Exercises 1-4, find the volume under the surface z= f(x,y) over the rectangle R. 1. f(x,y)=4xy, R=[0,1]×[0,1] 2. f(x,y)=e x+y , R=[0,1]×[−1,1] 3. f(x,y)= x 3 +y 2 , R=[0,1]×[0,1] 4. f(x,y)= x 4 + xy+y 3 , R=[1,2]×[0,2] For Exercises 5-12, evaluate the given double integral. 5. 7. 9. 11. ∫ 1 ∫ 2 0 1 ∫ 2 ∫ 1 0 0 ∫ π/2 ∫ 1 0 0 ∫ 2 ∫ 4 0 1 (1−y)x 2 dxdy 6. (x+2)dxdy 8. xycos(x 2 y)dxdy 10. xydxdy 12. 13. Let M be a constant. Show that ∫ d c ∫ b a ∫ 1 ∫ 2 0 0 ∫ 2 ∫ 1 −1 −1 ∫ π ∫ π/2 0 0 ∫ 1 ∫ 2 −1 −1 x(x+y)dxdy x(xy+sinx)dxdy sinxcos(y−π)dxdy 1dxdy Mdxdy= M(d−c)(b−a).
3.2 Double Integrals Over a General Region 105 3.2 Double Integrals Over a General Region In the previous section we got an idea of what a double integral over a rectangle represents. We can now define the double integral of a real-valued function f(x,y) over more general regions in 2 . Suppose that we have a region R in the xy-plane that is bounded on the left by the vertical line x=a, bounded on the right by the vertical line x=b (where a
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About the author: Michael Corral is
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iv Preface matter, anyone can make
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vi Contents 4.4 Surface Integrals a
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2 CHAPTER 1. VECTORS IN EUCLIDEAN S
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188 Bibliography Pogorelov, A.V., A
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190 Appendix A: Answers and Hints t
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Appendix B We will prove the right-
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Appendix C 3D Graphing with Gnuplot
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200 Appendix C: 3D Graphing with Gn
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202 GNU Free Documentation License
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Index Symbols D....................
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212 Index iterated integral .......
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Michael Corral: Vector Calculus

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