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CHAPTER 13
Multiple Integrals
In exercises 23-26, set up a double integral for the volume bounded by the given surfaces and estimate it numerically.
23. \( z=\sqrt{x^{2}+y^{2}}, y=4-x^{2} \), first octant
24. \( z=\sqrt{4-x^{2}-y^{2}} \), inside \( x^{2}+y^{2}=1 \), first octant
25. \( z=e^{x y}, x+2 y=4 \) and the three coordinate planes
26. \( z=e^{x^{2}+y^{2}}, z=0 \) and \( x^{2}+y^{2}=4 \)
In exercises 27-32, find the mass and center of mass of the lamina with the given density.
27. Lamina bounded by \( y=x^{3} \) and \( y=x^{2}, \rho(x, y)=4 \)
28. Lamina bounded by \( y=x^{4} \) and \( y=x^{2}, \rho(x, y)=4 \)
29. Lamina bounded by \( x=y^{2} \) and \( x=1, \rho(x, y)=y^{2}+x+1 \)
30. Lamina bounded by \( x=y^{2} \) and \( x=4, \rho(x, y)=y+3 \)
31. Lamina bounded by \( y=x^{2}(x>0), y=4 \) and \( x=0 \), \( \rho(x, y)= \) distance from \( y \)-axis
32. Lamina bounded by \( y=x^{2}-4 \) and \( y=5, \rho(x, y)= \) square of the distance from the \( y \)-axis
33. (a) The laminae of exercises 29 and 30 are both symmetric about the \( x \)-axis. Explain why it is not true in both exercises that the center of mass is located on the \( x \)-axis. (b) Suppose that a lamina is symmetric about the \( x \)-axis. State a condition on the density function \( \rho(x, y) \) that guarantees that the center of mass is located on the \( x \)-axis.
34. Suppose that a lamina is symmetric about the \( y \)-axis. State a
