Chapter Review

THINGS TO KNOW

14.1 The Double Integral over a Rectangular Region

14.2 The Double Integral over Nonrectangular Regions

14.3 Double Integrals Using Polar Coordinates

969

14.4 Center of Mass; Moment of Inertia

14.5 Surface Area

14.6 The Triple Integral

14.7 Triple Integrals Using Cylindrical Coordinates

14.8 Triple Integrals Using Spherical Coordinates

14.9 Change of Variables Using Jacobians

OBJECTIVES

Section You should be able to … Example Review Exercises
14.1 1 Find Riemann sums of \(z=f(x, y)\) over a closed rectangular region (p. 904) 1 7(a), 7(b)
2 Find the value of a double integral defined on a closed rectangular region (p. 906) 2, 3, 4 1, 2, 7(c), 8, 9
3 Find the volume under a surface and over a rectangular region (p. 908) 5 10, 11
14.2 1 Use Fubini’s Theorem for an \(x\)-simple region (p. 914) 1 3–6, 13
2 Use Fubini’s Theorem for a \(y\)-simple region (p. 916) 2, 3 3–6, 13, 14, 54
3 Work with properties of double integrals (p. 917) 4, 5 12
4 Use double integrals to find area and volume (p. 918) 6, 7, 8 15–18, 20
14.3 1 Find a double integral using polar coordinates (p. 924) 1, 2, 3 19, 52
2 Find area and volume using polar coordinates (p. 925) 4, 5, 6 21, 22
14.4 1 Find the mass and center of mass of a lamina (p. 929) 1, 2 23–25
2 Find moments of inertia (p. 933) 3 26
14.5 1 Find the surface area that lies above a region \(R\) (p. 938) 1, 2, 3 27–29
14.6 1 Find a triple integral defined over a closed box (p. 942) 1 30
2 Find a triple integral defined over a more general solid (p. 942) 2 31, 32
3 Find the volume of a solid (p. 944) 3 15–17, 35, 36
4 Find the mass, center of mass, and moments of inertia of a solid (p. 944) 4, 5 62
5 Find a triple integral defined over \(xz\)-simple and \(yz\)-simple solids (p. 946) 6 33
14.7 1 Convert rectangular coordinates to cylindrical coordinates (p. 950) 1 37(a)–40(a)
2 Find a triple integral using cylindrical coordinates (p. 951) 2, 3, 4, 5 43–45, 47–51, 53
14.8 1 Convert rectangular coordinates to spherical coordinates (p. 956) 1, 2 37(b)–40(b)
2 Find a triple integral using spherical coordinates (p. 958) 3, 4, 5, 6 34, 41, 42, 46–51
14.9 1 Find a Jacobian in two variables (p. 963) 1 55, 56
2 Change the variables of a double integral using a Jacobian (p. 964) 2, 3 58–60
3 Change the variables of a triple integral using a Jacobian (p. 965) 4 57, 61