1 Functions And Models 2 Limits And Derivatives 3 Differentiation Rules 4 Applications Of Differentiation 5 Integrals 6 Applications Of Integration 7 Techniques Of Integration 8 Further Applications Of Integration 9 Differential Equations 10 Parametric Equations And Polar Coordinates 11 Sequences, Series, And Power Series 12 Vectors And The Geometry Of Space 13 Vector Functions 14 Partial Derivatives 15 Multiple Integrals 16 Vector Calculus A Numbers, Inequalities, And Absolute Values B Coordinate Geometry And Lines C Graphs Of Second-degree Equations D Trigonometry E Sigma Notation F Proofs Of Theorems G The Logarithm Defined As An Integral expand_more
15.1 Double Integrals Over Rectangles 15.2 Double Integrals Over General Regions 15.3 Double Integrals In Polar Coordinates 15.4 Applications Of Double Integrals 15.5 Surface Area 15.6 Triple Integrals 15.7 Triple Integrals In Cylindrical Coordinates 15.8 Triple Integrals In Spherical Coordinates 15.9 Change Of Variables In Multiple Integrals Chapter Questions expand_more
Problem 1CC Problem 2CC Problem 3CC: How do you change from rectangular coordinates to polar coordinates in a double integral? Why would... Problem 4CC: If a lamina occupies a plane region D and has density function (x, y), write expressions for each of... Problem 5CC Problem 6CC Problem 7CC Problem 8CC Problem 9CC Problem 10CC Problem 1TFQ Problem 2TFQ Problem 3TFQ Problem 4TFQ Problem 5TFQ Problem 6TFQ: Determine whether the statement is true or false. If it is true, explain why. If it is false,... Problem 7TFQ: Determine whether the statement is true or false. If it is true, explain why. If it is false,... Problem 8TFQ Problem 9TFQ Problem 1E: A contour map is shown for a function f on the square R = [0, 3] [0, 31. Use a Riemann sum with... Problem 2E: Use the Midpoint Rule to estimate the integral in Exercise 1. 1. A contour map is shown for a... Problem 3E: Calculate the iterated integral. 3. 1202(y+2xey)dxdy Problem 4E: Calculate the iterated integral. 4. 0101yexydxdy Problem 5E: Calculate the iterated integral. 5. 010xcos(x2)dydx Problem 6E: Calculate the iterated integral. 6. 01xex3xy2dydx Problem 7E: Calculate the iterated integral. 7. 00101y2ysinxdzdydx Problem 8E: Calculate the iterated integral. 8. 010yx16xyzdzdxdy Problem 9E: Write Rf(x,y)dA as an iterated integral, where R is the region shown and f is an arbitrary... Problem 10E: Write Rf(x,y)dA as an iterated integral, where R is the region shown and f is an arbitrary... Problem 11E: The cylindrical coordinates of a point are (23,3, 2). Find the rectangular and spherical coordinates... Problem 12E Problem 13E: The spherical coordinates of a point are (8, /4, /6). Find the rectangular and cylindrical... Problem 14E: Identify the surfaces whose equations are given. (a) = /4 (b) = /4 Problem 15E: Write the equation in cylindrical coordinates and in spherical coordinates. (a) x2 + y2 + z2 = 4 (b)... Problem 16E Problem 17E: Describe the region whose area is given by the integral 0/20sin2rdrd Problem 18E: Describe the solid whose volume is given by the integral 0/20/2122sinddd and evaluate the integral. Problem 19E: Calculate the iterated integral by first reversing the order of integration. 01x1cos(y2)dydx Problem 20E: Calculate the iterated integral by first reversing the order of integration. 01y1yex2x3dxdy Problem 21E: Calculate the value of the multiple integral. 21. RyexydA, where R = {(x, y) | 0 x 2, 0 y 3} Problem 22E: Calculate the value of the multiple integral. 22. DxydA, where D = {(x, y) | 0 y 1, y2 x y + 2} Problem 23E: Calculate the value of the multiple integral. 23. Dy1+x2dA, where D is bounded by y=x, y = 0, x = 1 Problem 24E: Calculate the value of the multiple integral. 24. Dy1+x2dA, where D is the triangular region with... Problem 25E: Calculate the value of the multiple integral. 25. DydA, where D is the region in the first quadrant... Problem 26E: Calculate the value of the multiple integral. 26. DydA, where D is the region in the first quadrant... Problem 27E: Calculate the value of the multiple integral. 27. D(x2+y2)3/2dA,where /9 is the region in the first... Problem 28E: Calculate the value of the multiple integral. 28. DxdA, where D is the region in the first quadrant... Problem 29E: Calculate the value of the multiple integral. 29. ExydV, where E = {(x, y, z) | 0 x 3, 0 y x, 0 ... Problem 30E Problem 31E: Calculate the value of the multiple integral. 31. Ey2z2dV, where E is bounded by the paraboloid x =... Problem 32E: Calculate the value of the multiple integral. 32. EzdV, where E is bounded by the planes y = 0, z =... Problem 33E: Calculate the value of the multiple integral. 33. EyzdV, where E lies above the plane z = 0, below... Problem 34E Problem 35E Problem 36E Problem 37E Problem 38E Problem 39E Problem 40E Problem 41E: Consider a lamina that occupies the region D bounded by the parabola x = 1 y2 and the coordinate... Problem 42E: A lamina occupies the part of the disk x2 + y2 a2 that lies in the first quadrant. (a) Find the... Problem 43E: (a) Find the centroid of a solid right circular cone with height hand base radius a. (Place the cone... Problem 44E: Find the area of the part of the cone z2 = a2(x2 + y2) between the planes z = 1 and z = 2. Problem 45E Problem 46E Problem 47E: Use polar coordinates to evaluate 039x29x2(x3+xy2)dydx Problem 48E: Use spherical coordinates to evaluate 2204y24x2y24x2y2y2x2+y2+z2dzdxdy Problem 49E Problem 50E Problem 51E Problem 52E: A lamp has three bulbs, each of a type with average lifetime 800 hours. If we model the probability... Problem 53E Problem 54E Problem 55E Problem 56E: Use the transformation x = u2, y = v2 z = w2 to find the volume of the region bounded by the surface... Problem 57E Problem 58E Problem 1PP Problem 2PP: Evaluate the integral 0101emaxx2,y2dydxwhere max{x2, y2} means the larger of the numbers x2 and y2. Problem 3PP Problem 4PP Problem 5PP: The double integral 010111xydxdyis an improper integral and could be defined as the limit of double... Problem 6PP: Leonhard Euler was able to find the exact sum of the series in Problem 5. In 1736 he proved... Problem 7PP Problem 8PP Problem 9PP: (a) Show that when Laplaces equation 2ux2+2uy2+2uz2=0is written in cylindrical coordinates, it... Problem 10PP: (a) A lamina has constant density and takes the shape of a disk with center the origin and radius... Problem 11PP: If f is continuous, show that 0x0y0zf(t)dtdzdy=120x(xt)2f(t)dt Problem 12PP: Evaluate limnn2i=1nj=1n21n2+ni+j. Problem 13PP: The plane xa+yb+zc=1a0,b0,c0cuts the solid ellipsoid x2a2+y2b2+z2c21 into two pieces. Find the... format_list_bulleted
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