 12.12.5.1: Evaluate the integral in Example 1, integrating first with respect ...
 12.12.6.1: Plot the point whose cylindrical coordinates are given. Then find t...
 12.12.1.1: (a) Estimate the volume of the solid that lies below m n 2 the surf...
 12.1: Suppose is a continuous function defined on a rectangle . (a) Write...
 12.12.2.1: Evaluate the iterated integral.0 y x 2 0 x 2y dy dx Pr
 12.12.3.1: A region is shown. Decide whether to use polar coordinates or recta...
 12.12.4.1: Electric charge is distributed over the rectangle 13. , so that the...
 12.12.5.2: Evaluate the integral , where using three different orders of integ...
 12.12.6.2: Plot the point whose cylindrical coordinates are given. Then find t...
 12.12.1.2: If , use a Riemann sum with , to estimate the value of . Take the s...
 12.2: (a) How do you define if is a bounded region that is not a rectangl...
 12.12.2.2: Evaluate the iterated integral.2 1 y 2 y xy dx dy
 12.12.3.2: A region is shown. Decide whether to use polar coordinates or recta...
 12.12.4.2: Electric charge is distributed over the disk so that the charge den...
 12.12.4.3: Find the mass and center of mass of the lamina that occupies the re...
 12.12.5.3: Evaluate the iterated integral.y 2xyz dz dy dx 1 0 y z 0 y xz 0 6xz...
 12.12.6.3: Change from rectangular to cylindrical coordinates.3. 1, 1, 4 (1, s...
 12.12.7.3: Change from rectangular to spherical coordinates.(1, s3 , 2s3 ) 0, ...
 12.12.1.3: (a) Use a Riemann sum with to estimate the value of , where . Take ...
 12.3: How do you change from rectangular coordinates to polar coordinates...
 12.12.2.3: Evaluate the iterated integral.1 0 y e y y sx dx dy
 12.12.3.3: A region is shown. Decide whether to use polar coordinates or recta...
 12.12.4.4: Find the mass and center of mass of the lamina that occupies the re...
 12.12.5.4: Evaluate the iterated integral.y 1 0 y 2x x y y 0 y 2xyz dz dy dx
 12.12.6.4: Change from rectangular to cylindrical coordinates.3, 3, 2 3, 4, 5 ...
 12.12.7.4: Change from rectangular to spherical coordinates.(0, s3 , 1) (1, 1,...
 12.12.1.4: (a) Estimate the volume of the solid that lies below the surface an...
 12.4: If a lamina occupies a plane region and has density function , writ...
 12.12.2.4: Evaluate the iterated integral.1 0 y 2x x x y 2 y dy dx 1 0
 12.12.3.4: A region is shown. Decide whether to use polar coordinates or recta...
 12.12.4.5: Find the mass and center of mass of the lamina that occupies the re...
 12.12.5.5: Evaluate the iterated integral.y dx dy dz 3 0 y 1 0 y s1z 2 0 zey d...
 12.12.6.5: Describe in words the surface whose equation is given.r 3
 12.12.7.5: Describe in words the surface whose equation is given.3
 12.12.1.5: A contour map is shown for a function on the square . Use the Midpo...
 12.5: (a) Write the definition of the triple integral of over a rectangul...
 12.12.2.5: Evaluate the iterated integral.y du dv 2 0 y cos 0 esin 5. dr d
 12.12.3.5: Sketch the region whose area is given by the integral and evaluate ...
 12.12.4.6: Find the mass and center of mass of the lamina that occupies the re...
 12.12.5.6: Evaluate the iterated integral.y 1 0 y z 0 y y 0 zey 2 y dx dy dz 3
 12.12.6.6: Describe in words the surface whose equation is given.3
 12.12.7.6: Describe in words the surface whose equation is given.3
 12.12.1.6: A 20ftby30ft swimming pool is filled with water. The depth is m...
 12.6: Suppose a solid object occupies the region and has density function...
 12.12.2.6: Evaluate the iterated integral.y 1 0 y v 0 s1 v 2 y du dv
 12.12.3.6: Sketch the region whose area is given by the integral and evaluate ...
 12.12.4.7: Find the mass and center of mass of the lamina that occupies the re...
 12.12.6.7: Identify the surface whose equation is given.z r 2 4 2 r
 12.12.7.7: Identify the surface whose equation is given.sin 2
 12.12.8.7: Find the image of the set under the given transformation.u, v 7. 0 ...
 12.12.1.7: Evaluate the double integral by first identifying it as the volume ...
 12.7: (a) Write the equations for converting from cylindrical to rectangu...
 12.12.2.7: Evaluate the double integral.x 3 y 2 dA, D x, y 0 x 2, x y x 16
 12.12.3.7: xxD xy dA where is the disk with center the origin and radius 3
 12.12.4.8: Find the mass and center of mass of the lamina that occupies the re...
 12.12.5.8: Evaluate the triple integral.E x, y, z 0 x 1, 0 y x, x z 2x xx
 12.12.6.8: Identify the surface whose equation is given.r 2 2z z r 2 4 2
 12.12.7.8: Identify the surface whose equation is given.2 cos
 12.12.8.8: Find the image of the set under the given transformation.8. is the ...
 12.12.1.8: Evaluate the double integral by first identifying it as the volume ...
 12.8: (a) How do you change from rectangular coordinates to cylindrical c...
 12.12.2.8: Evaluate the double integral.yy D 4y x 3 2 dA, D x, y 1 x 2, 0 y 2x yy
 12.12.3.8: xx R R x y dA D here is the region that lies to the left of the ax...
 12.12.4.9: Find the mass and center of mass of the lamina that occupies the re...
 12.12.5.9: Evaluate the triple integral.xxx E z 1 x y E 9. 6xy dV E lies under...
 12.12.6.9: Write the equations in cylindrical coordinates.x 2 y z x 2 2y 2 y 2...
 12.12.7.9: Write the equation in spherical coordinates.x 2 y z x 2 2y 2 y 2 si
 12.12.8.9: Find the image of the set under the given transformation.9. is the ...
 12.12.1.9: Evaluate the double integral by first identifying it as the volume ...
 12.9: (a) If a transformation is given by , what is the Jacobian of ? (b)...
 12.12.2.9: Evaluate the double integral.yy D 2y x 2 1 dA, D {x, y 0 x 1, 0 y s...
 12.12.3.9: xx R R cosx 2 y 2 dA x where is the region that lies above the axi...
 12.12.4.10: Find the mass and center of mass of the lamina that occupies the re...
 12.12.5.10: Evaluate the triple integral.xxx E x 0 y 0 E y dV y where is bounde...
 12.12.6.10: Write the equations in cylindrical coordinates.z x 2 y 2 x 2 y 2 z ...
 12.12.7.10: Write the equation in spherical coordinates.z x 2 y 2 x 2 y 2 z 2 2...
 12.12.8.10: Find the image of the set under the given transformation.10. is the...
 12.12.1.10: The integral , where , represents the volume of a solid. Sketch the...
 12.10: Write as an iterated integral, where is the region shown and f is a...
 12.12.2.10: Evaluate the double integral.yy D 2y x 2 1 dA, D {x, y 0 x 1, 0 y s...
 12.12.3.10: xxR s4 x 2 y 2 dA x where R x, y x 2 y 2 4, x 0 xxR
 12.12.4.11: A lamina occupies the part of the disk in the first quadrant. Find ...
 12.12.5.11: Evaluate the triple integral. xxx E E xy dV where is the solid tetr...
 12.12.6.11: Sketch the solid described by the given inequalities.r 2 z 2 r 2 z
 12.12.8.11: xx R R x 3y dA u where is the triangular region with vertices ,0, 0...
 12.12.1.11: Calculate the iterated integral.y dy dx 3 1 y 1 0 11. 1 4xy dx dy xx
 12.11: Describe the region whose area is given by the integral 2 0 y sin 2...
 12.12.2.11: Evaluate the double integral.y x x 1 2 yy D y 0 D
 12.12.3.11: xxR 11. arctan yx dA R where R = x, y 1 x 2 y 2 4, 0 y x xxR
 12.12.4.12: Find the center of mass of the lamina in Exercise 11 if the density...
 12.12.5.12: Evaluate the triple integral. xx E E xz dV where is the solid tetra...
 12.12.6.12: Sketch the solid described by the given inequalities.0 2 r z 2
 12.12.8.12: xx R R 4x 8y dA 0, where is the parallelogram with vertices , , , a...
 12.12.1.12: Calculate the iterated integral.y 4 2 y 1 1 x 2 y 2 y dy dx 3 1
 12.12: Describe the solid whose volume is given by the integral and evalua...
 12.12.2.12: Evaluate the double integral.y sx and y x 2 yy D D
 12.12.3.12: xx R R yex dA where is the region in the first quadrant enclosed by...
 12.12.4.13: Find the center of mass of a lamina in the shape of an isosceles ri...
 12.12.5.13: Evaluate the triple integral. xxx E E x 2 ey dV where is bounded by...
 12.12.6.13: A cylindrical shell is 20 cm long, with inner radius 6 cm and outer...
 12.12.7.13: Sketch the solid described by the given inequalities.2 2 0 6 0 sec 2
 12.12.8.13: x R R x 2 13. dA , where is the region bounded by the ellipse 9x x ...
 12.12.1.13: Calculate the iterated integral.0 y 2 0 x sin y dy dx
 12.13: Calculate the iterated integral by first reversing the order of int...
 12.12.2.13: Evaluate the double integral.yy D y 3 dA is the triangular region w...
 12.12.3.13: Under the cone and above the disk
 12.12.4.14: A lamina occupies the region inside the circle but outside the circ...
 12.12.5.14: Evaluate the triple integral. xxx E E x 2y dV z where is bounded by...
 12.12.6.14: Use a graphing device to draw the solid enclosed by theparaboloids ...
 12.12.7.14: Sketch the solid described by the given inequalities.0 3 2
 12.12.8.14: xx R R x 2 xy y 2 dA 9 14. , where is the region bounded by the ell...
 12.12.1.14: Calculate the iterated integral.y 4 1 y 2 0 y (x sy ) dx dy 2
 12.14: Calculate the iterated integral by first reversing the order of int...
 12.12.2.14: Evaluate the double integral.yy D xy 2 dA, D is enclosed by x 0 and...
 12.12.3.14: Below the paraboloid and above the plane
 12.12.4.15: Find the moments of inertia , , for the lamina of Exercise 7.
 12.12.5.15: Evaluate the triple integral. xxx E E x dV is bounded by the parabo...
 12.12.6.15: Sketch the solid whose volume is given by the integral and evaluate...
 12.12.7.15: A solid lies above the cone and below the sphere . Write a descript...
 12.12.8.15: where is the region in the first quadrant bounded by the lines and ...
 12.12.1.15: Calculate the iterated integral.y dy dx 2 0 y 1 0 2x y 8 dx dy y
 12.15: x, y 0 y 3 xx 0 x 2 R yexy dA y
 12.12.2.15: Evaluate the double integral.yy D 15. 2x y dA, yy is bounded by the...
 12.12.3.15: A sphere of radius a
 12.12.4.16: Find the moments of inertia , , for the lamina of Exercise 12.
 12.12.5.16: Evaluate the triple integral. xxx E 2 9 E z dV where is bounded by ...
 12.12.6.16: Sketch the solid whose volume is given by the integral and evaluate...
 12.12.7.16: (a) Find inequalities that describe a hollow ball with diameter 30 ...
 12.12.8.16: xx R R y 2 16 dA ; 16. , where is the region bounded by the curves ...
 12.12.1.16: Calculate the iterated integral.y 1 0 y 2 1 xex y y dy dx 2
 12.16: x, y 2 x y 2 xx 0 y 1 D xy dA R
 12.12.2.16: Evaluate the double integral.yy 0, 0 D 2xy dA, D D 16. is the trian...
 12.12.3.16: Inside the sphere and outside the cylinder
 12.12.4.17: Find the moments of inertia , , for the lamina of Exercise 9.
 12.12.5.17: The tetrahedron enclosed by the coordinate planes and the plane
 12.12.6.17: Evaluate , where is the region that lies inside the cylinder and be...
 12.12.7.17: Sketch the solid whose volume is given by the integral and evaluate...
 12.12.8.17: . (a) Evaluate , where is the solid enclosed by the ellipsoid . Use...
 12.12.1.17: Calculate the iterated integral.y dx dy 4 1 y 2 1 x y y x 17. dy dx y
 12.17: yy D y 1 x 2 dA where is bounded by , , y sx y 0 x 1 yy
 12.12.2.17: Find the volume of the given solid.Under the plane and above the re...
 12.12.3.17: Above the cone and below the sphere
 12.12.4.18: Consider a square fan blade with sides of length 2 and the lower le...
 12.12.5.18: The solid bounded by the cylinder and the planes , and
 12.12.6.18: Evaluate , where is the solid in the first octant that lies beneath...
 12.12.7.18: Sketch the solid whose volume is given by the integral and evaluate...
 12.12.8.18: Evaluate , where is the solid of Exercise 17(a).
 12.12.1.18: Calculate the iterated integral.y 2 1 y 1 0 x y 2 y dx dy 4 1
 12.18: yy D D 1 1 x 2 dA 18. , where is the triangular region with 0, 0 1,...
 12.12.2.18: Find the volume of the given solid.Under the surface and above the ...
 12.12.3.18: Bounded by the paraboloid and the plane in the first quadrant
 12.12.4.19: x, y x, y xy 0 y sin x, 0 x D CA
 12.12.5.19: The solid enclosed by the cylinder and the planes and
 12.12.6.19: Evaluate , where is enclosed by the paraboloid , the cylinder , and...
 12.12.7.19: Set up the triple integral of an arbitrary continuous function in c...
 12.12.8.19: yy R R x 2y 3x y dA x . , where is the parallelogram enclosed by x ...
 12.12.1.19: Calculate the iterated integral.y dy dx ln 2 0 y ln 5 0 e 2xy dx dy
 12.19: xxD y dA 19. , where is the region in the first quadrant bounded by...
 12.12.2.19: Find the volume of the given solid.Under the surface and above the ...
 12.12.3.19: Inside both the cylinder and the ellipsoid 4x 2 4y 2 z2 64 x 2
 12.12.4.20: D is enclosed by the cardioid ; x, y sx 2 y 2 D r
 12.12.5.20: The solid enclosed by the paraboloid and the plane
 12.12.6.20: Evaluate , where is enclosed by the planes and and by the cylinders...
 12.12.7.20: Set up the triple integral of an arbitrary continuous function in c...
 12.12.8.20: . , where is the rectangle enclosed by x y 0 x y 2 x y 0 x y 3 xx R...
 12.12.1.20: Calculate the iterated integral.y 1 0 y 1 0 xysx 2 y 2 y dy dx
 12.20: xxD y dA D 20. , where is the region in the first quadrant that lie...
 12.12.2.20: Find the volume of the given solid.Enclosed by the paraboloid and t...
 12.12.3.20: (a) A cylindrical drill with radius is used to bore a hole through ...
 12.12.4.21: A lamina with constant density occupies a square with vertices , , ...
 12.12.5.21: (a) Express the volume of the wedge in the first octant that is cut...
 12.12.6.21: Evaluate , where is the solid that lies within the cylinder , above...
 12.12.7.21: Use spherical coordinates.Evaluate , where is the unit ball x 2 y 2...
 12.12.8.21: yy R R cos y x y x 21. dA x , where is the trapezoidal region with ...
 12.12.1.21: Calculate the double integral.xy 2 x 2 1 dA R x, y yy 0 x 1, 3 y 3 R
 12.21: xxD x D 2 y 2 32 21. , where is the region in the first quadrant bo...
 12.12.2.21: Find the volume of the given solid.Bounded by the planes , , , and ...
 12.12.3.21: Use a double integral to find the area of the region.One loop of th...
 12.12.4.22: A lamina with constant density occupies the region under the curve ...
 12.12.5.22: (a) In the Midpoint Rule for triple integrals we use a triple Riema...
 12.12.6.22: Find the volume of the solid that lies within both the cylinder and...
 12.12.7.22: Use spherical coordinates.Evaluate , where is the hemispherical reg...
 12.12.8.22: xx R R sin9x 2 4y 2 dA 1, 22. , where is the region in the firstqua...
 12.12.1.22: Calculate the double integral.yy R cosx 2y dA R R x, y 0 x , 0 y 2 yy
 12.22: xxD x dA D 22. , where is the region in the first quadrant that lie...
 12.12.2.22: Find the volume of the given solid.Bounded by the planes , , , and
 12.12.3.22: Use a double integral to find the area of the region.The region enc...
 12.12.5.23: Use the Midpoint Rule for triple integrals (Exercise 22) to estimat...
 12.12.6.23: (a) Find the volume of the region bounded by the paraboloids and . ...
 12.12.7.23: Use spherical coordinates.Evaluate , where lies between the spheres...
 12.12.8.23: xx R R exy dA 23. , where is given by the inequality x y 1 x
 12.12.1.23: Calculate the double integral.yy R 0, 6 0, 3
 12.23: xxxE xy dV x, y, z 0 y x 0 z x y 0 x 3 xxx
 12.12.2.23: Find the volume of the given solid.Enclosed by the cylinders , and ...
 12.12.3.23: Evaluate the iterated integral by converting to polar coordinates.y...
 12.12.5.24: Use the Midpoint Rule for triple integrals (Exercise 22) to estimat...
 12.12.6.24: (a) Find the volume of the solid that the cylinder cuts out of the ...
 12.12.7.24: Use spherical coordinates.Evaluate , where is enclosed by the spher...
 12.12.8.24: Let be continuous on and let be the triangular region with vertices...
 12.12.1.24: Calculate the double integral.yy 0 x 1, 0 y 1 R 1 x 2 1 y 2 dA y
 12.24: xxx T T xy dV 24. , where is the solid tetrahedron with vertices 3 ...
 12.12.2.24: Find the volume of the given solid.Bounded by the cylinder and the ...
 12.12.3.24: Evaluate the iterated integral by converting to polar coordinates.y...
 12.12.5.25: Sketch the solid whose volume is given by the iterated integral.y 1...
 12.12.6.25: Find the mass and center of mass of the solid bounded by the parabo...
 12.12.7.25: Use spherical coordinates. Evaluate , where is bounded by the plan...
 12.12.1.25: Calculate the double integral.y , R 0, 1 0, 2 R xye x2y dA
 12.25: xxx E E y 2 25. , where is bounded by the paraboloid x 1 y x 0 2 z2...
 12.12.2.25: Find the volume of the given solid.Bounded by the cylinder and the ...
 12.12.3.25: Evaluate the iterated integral by converting to polar coordinates.y...
 12.12.5.26: Sketch the solid whose volume is given by the iterated integral.2 0...
 12.12.6.26: Find the mass of a ball given by if the P density at any point is p...
 12.12.7.26: Use spherical coordinates.Find the volume of the solid that lies wi...
 12.12.1.26: Calculate the double integral.yy R 0, 1 0, 1 R x 1 xy dA 6.
 12.26: xxx E y 0 z 0 E z dV x 26. , where is bounded by the planes , , and...
 12.12.2.26: Find the volume of the given solid.Bounded by the cylinders and
 12.12.3.26: Evaluate the iterated integral by converting to polar coordinates.y...
 12.12.5.27: Express the integral as an iterated integral in six different ways,...
 12.12.6.27: Evaluate the integral by changing to cylindrical coordinates.y 2 2 ...
 12.12.7.27: Use spherical coordinates.(a) Find the volume of the solid that lie...
 12.12.1.27: Find the volume of the solid that lies under the plane and above th...
 12.27: xxx E z 0 E yz dV y 27. , where lies above the plane , below the pl...
 12.12.2.27: Find the volume of the solid by subtracting two volumes.The solid e...
 12.12.3.27: A swimming pool is circular with a 40ft diameter. The depth is con...
 12.12.5.28: Express the integral as an iterated integral in six different ways,...
 12.12.6.28: Evaluate the integral by changing to cylindrical coordinates.3 3 y ...
 12.12.7.28: Let be a solid hemisphere of radius whose density at any point is p...
 12.12.1.28: Find the volume of the solid that lies under the hyperbolic parabol...
 12.28: xxxH z H 3 sx 2 y 2 z 2 dV x 28. , where is the solid hemisphere th...
 12.12.2.28: Find the volume of the solid by subtracting two volumes.The solid e...
 12.12.3.28: An agricultural sprinkler distributes water in a circular pattern o...
 12.12.5.29: Express the integral as an iterated integral in six different ways,...
 12.12.6.29: When studying the formation of mountain ranges, geologists estimate...
 12.12.7.29: Use spherical coordinates.(a) Find the centroid of a solid homogene...
 12.12.1.29: Find the volume of the solid lying under the elliptic paraboloid an...
 12.29: Find the volume of the given solid. Under the paraboloid and above ...
 12.12.2.29: Enclosed by z 1 x 2 y 2 an and z 0 C
 12.12.3.29: Use polar coordinates to combine the sum into one double integral. ...
 12.12.5.30: Express the integral as an iterated integral in six different ways,...
 12.12.7.30: Use spherical coordinates.Find the mass and center of mass of a sol...
 12.12.1.30: Find the volume of the solid enclosed by the surface and the planes...
 12.30: Find the volume of the given solid.Under the surface and above the ...
 12.12.2.30: Enclosed by z x 2 y 2 a and z 2y z
 12.12.3.30: (a) We define the improper integral (over the entire plane where is...
 12.12.5.31: The figure shows the region of integration for the integral Rewrite...
 12.12.7.31: Find the volume and centroid of the solid that lies above the cone ...
 12.12.1.31: Find the volume of the solid bounded by the surface and the planes ...
 12.31: Find the volume of the given solid. The solid tetrahedron with vert...
 12.12.2.31: Sketch the region of integration and change the order of integratio...
 12.12.3.31: Use the result of Exercise 30 part (c) to evaluate the following in...
 12.12.5.32: The figure shows the region of integration for the integral CAS E R...
 12.12.7.32: Find the volume of the smaller wedge cut from a sphere of radius by...
 12.12.1.32: Find the volume of the solid bounded by the elliptic paraboloid , t...
 12.32: Find the volume of the given solid.Bounded by the cylinder and the ...
 12.12.2.32: Sketch the region of integration and change the order of integratio...
 12.12.5.33: Write five other iterated integrals that are equal to the given ite...
 12.12.7.33: Evaluate , where lies above the paraboloid and below the plane . Us...
 12.12.1.33: Find the volume of the solid in the first octant bounded by the cyl...
 12.33: Find the volume of the given solid.One of the wedges cut from the c...
 12.12.2.33: Sketch the region of integration and change the order of integratio...
 12.12.5.34: Write five other iterated integrals that are equal to the given ite...
 12.12.7.34: (a) Find the volume enclosed by the torus . ; (b) Use a computer to...
 12.12.1.34: (a) Find the volume of the solid bounded by the surface and the pla...
 12.34: Find the volume of the given solid.Above the paraboloid and below t...
 12.12.2.34: Sketch the region of integration and change the order of integratio...
 12.12.5.35: Find the mass and center of mass of the solid with the given densit...
 12.12.7.35: Evaluate the integral by changing to spherical coordinates.y 1 0 y ...
 12.12.1.35: Use a computer algebra system to find the exact value of the integr...
 12.35: . Consider a lamina that occupies the region bounded by the parabol...
 12.12.2.35: Sketch the region of integration and change the order of integratio...
 12.12.5.36: Find the mass and center of mass of the solid with the given densit...
 12.12.7.36: Evaluate the integral by changing to spherical coordinates.y a a y ...
 12.12.1.36: Graph the solid that lies between the surfaces and for , . Use a co...
 12.36: A lamina occupies the part of the disk that lies in the first quadr...
 12.12.2.36: Sketch the region of integration and change the order of integratio...
 12.12.5.37: Find the mass and center of mass of the solid with the given densit...
 12.12.7.37: Use a graphing device to draw a silo consisting of a cylinder with ...
 12.12.1.37: The average value of a function over a rectangle is defined to be (...
 12.37: (a) Find the centroid of a right circular cone with height and base...
 12.12.2.37: Evaluate the integral by reversing the order of integration.y 3 1 d...
 12.12.5.38: Find the mass and center of mass of the solid with the given densit...
 12.12.7.38: The latitude and longitude of a point in the Northern Hemisphere ar...
 12.12.1.38: The average value of a function over a rectangle is defined to be (...
 12.38: Find the center of mass of the solid tetrahedron with vertices , , ...
 12.12.2.38: Evaluate the integral by reversing the order of integration.y 1 0 y...
 12.12.5.39: Set up, but do not evaluate, integral expressions for (a) the mass,...
 12.12.7.39: The surfaces have been used as models for tumors. The bumpy sphere ...
 12.12.1.39: If is a constant function, , and , show that
 12.39: The cylindrical coordinates of a point are . Find the rectangular a...
 12.12.2.39: Evaluate the integral by reversing the order of integration.3 0 y 9...
 12.12.5.40: Set up, but do not evaluate, integral expressions for (a) the mass,...
 12.12.7.40: Show that (The improper triple integral is defined as the limit of ...
 12.12.1.40: If , show that
 12.40: The rectangular coordinates of a point are . Find the cylindrical a...
 12.12.2.40: Evaluate the integral by reversing the order of integration.y 1 0 y...
 12.12.5.41: Let be the solid in the first octant bounded by the cylinder and th...
 12.12.7.41: (a) Use cylindrical coordinates to show that the volume of the soli...
 12.12.1.41: Use your CAS to compute the iterated integrals Do the answers contr...
 12.41: The spherical coordinates of a point are . Find the rectangular and...
 12.12.2.41: Evaluate the integral by reversing the order of integration.y 1 0 y...
 12.12.5.42: If is the solid of Exercise 16 with density function , find the fol...
 12.12.1.42: (a) In what way are the theorems of Fubini and Clairaut similar? (b...
 12.42: Identify the surfaces whose equations are given. (a) (b)
 12.12.2.42: Evaluate the integral by reversing the order of integration.y 8 0 y...
 12.12.5.43: Find the moments of inertia for a cube of constant density and side...
 12.43: Write the equation in cylindrical coordinates and in spherical coor...
 12.12.2.43: Express as a union of regions of type I or type II and evaluate the...
 12.12.5.44: Find the moments of inertia for a rectangular brick with dimensions...
 12.44: Sketch the solid consisting of all points with spherical coordinate...
 12.12.2.44: Express as a union of regions of type I or type II and evaluate the...
 12.12.5.45: The average value of a function over a solid region is defined to b...
 12.45: Use polar coordinates to evaluate y 3 0 y s9x 2 s9x 2 x 3 xy 2 dy dx 0
 12.12.2.45: Use Property 11 to estimate the value of the integral.yy , D 0, 1 0, 1
 12.12.5.46: The average value of a function over a solid region is defined to b...
 12.46: Use spherical coordinates to evaluate2 2 y s4y 2 0 y s4x 2y 2 s4x 2...
 12.12.2.46: Use Property 11 to estimate the value of the integral.yy D ex 2 y 2...
 12.12.5.47: Find the region for which the triple integral is a maximum. yyy E 1...
 12.47: Rewrite the integral as an iterated integral in the order .1 1 y 1 ...
 12.12.2.47: Prove Property 11.
 12.48: Give five other iterated integrals that are equal to 2 0 y y 3 0 y ...
 12.12.2.48: In evaluating a double integral over a region , a sum of iterated i...
 12.49: Use the transformation , to evaluate , where is the square with ver...
 12.12.2.49: Evaluate , where [Hint: Exploit the fact that is symmetric with res...
 12.50: Use the transformation , , to find the volume of the region bounded...
 12.12.2.50: Use symmetry to evaluate , where is the region bounded by the squar...
 12.51: Use the change of variables formula and an appropriate transformati...
 12.12.2.51: Compute , where is the disk , by first identifying the integral as ...
 12.52: (a) Evaluate , where is an integer and is the region bounded by the...
 12.12.2.52: Graph the solid bounded by the plane and the paraboloid and find it...
Solutions for Chapter 12: MULTIPLE INTEGRALS
Full solutions for Essential Calculus (Available Titles CengageNOW)  1st Edition
ISBN: 9780495014423
Solutions for Chapter 12: MULTIPLE INTEGRALS
Get Full SolutionsSince 329 problems in chapter 12: MULTIPLE INTEGRALS have been answered, more than 16600 students have viewed full stepbystep solutions from this chapter. Chapter 12: MULTIPLE INTEGRALS includes 329 full stepbystep solutions. This textbook survival guide was created for the textbook: Essential Calculus (Available Titles CengageNOW), edition: 1. Essential Calculus (Available Titles CengageNOW) was written by and is associated to the ISBN: 9780495014423. This expansive textbook survival guide covers the following chapters and their solutions.

Complex conjugates
Complex numbers a + bi and a  bi

Head minus tail (HMT) rule
An arrow with initial point (x1, y1 ) and terminal point (x2, y2) represents the vector <8x 2  x 1, y2  y19>

Imaginary unit
The complex number.

Index of summation
See Summation notation.

Integrable over [a, b] Lba
ƒ1x2 dx exists.

Intermediate Value Theorem
If ƒ is a polynomial function and a < b , then ƒ assumes every value between ƒ(a) and ƒ(b).

Limit
limx:aƒ1x2 = L means that ƒ(x) gets arbitrarily close to L as x gets arbitrarily close (but not equal) to a

Normal curve
The graph of ƒ(x) = ex2/2

Parabola
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Permutation
An arrangement of elements of a set, in which order is important.

Probability function
A function P that assigns a real number to each outcome O in a sample space satisfying: 0 … P1O2 … 1, P12 = 0, and the sum of the probabilities of all outcomes is 1.

Pseudorandom numbers
Computergenerated numbers that can be used to approximate true randomness in scientific studies. Since they depend on iterative computer algorithms, they are not truly random

Semiperimeter of a triangle
Onehalf of the sum of the lengths of the sides of a triangle.

Stemplot (or stemandleaf plot)
An arrangement of a numerical data set into a specific tabular format.

Sum of an infinite geometric series
Sn = a 1  r , r 6 1

Symmetric matrix
A matrix A = [aij] with the property aij = aji for all i and j

Time plot
A line graph in which time is measured on the horizontal axis.

Trichotomy property
For real numbers a and b, exactly one of the following is true: a < b, a = b , or a > b.

Weighted mean
A mean calculated in such a way that some elements of the data set have higher weights (that is, are counted more strongly in determining the mean) than others.

xaxis
Usually the horizontal coordinate line in a Cartesian coordinate system with positive direction to the right,.