Math 251-copyright Joe Kahlig, 15A Page 11. True or FalseF x =RRDyρ(x, y)dARRDρ(x, y)dAT If a,b,c,d are constants, thenZbx=aZdy=cf(x, y)dydx =Zdy=cZbx=af(x, y)dxdyFZ20Z√4−x2−√4−x2g(x, y)dydx =Z20Zπ0g(r cos(θ), r sin(θ))rdθdrF The integralRRDf(x, y)dA is the volume under the the function f(x, y).F If a,b,c,d are constants, thenZbx=aZdy=c(x2+ y2)dydx =Zbx=ax2dxZdy=cy2dy2.π/2Z0sin(2θ)Z0r2cos2θ + 2r2sin2θr drdθ3. 6404.πZ02Z08−2r2Z−√4−r21rdzdrdθ5.4Z0√4−zZ0(2−x)/2Z0f(x, y, z)dydxdz6. Reverse the order of integration and then integrate. answer2993/2− 17. (a)π/2Z03Z0√36−r2Zr√3rzpr2+ z2dzdrdtheta(b) note: the sperical coordinates for the cone z2= 3x2+ 3y2is φ = π/6π/2Z0π/6Z06Z0ρ4cos φ sin φdρdφdθ8.6Z0(12−2x)/3Z0(12−2x−3y)/6Z01dzdydxMath 251-copyright Joe Kahlig, 15A Page 29. The equation of the plane z = 1 is ρ cos φ = 1 or ρ = sec φ.2πZ0π/4Z0√2Zsec ρρ2sin φdρdφdθ10.2πZ02Z1r cos θ+2Z0r2sin θdzdrdtheta11. The density at any point is proportional to the distance from the point (1, 1)ρ(x, y) = kp(x − 1)2+ (y − 1)2√2Z−√24−y2Zy2kp(x − 1)2+ (y − 1)2dxdy12. (a) top part of a cone centered on the z-axis. z =px2+ y2(b) a cylinder of radius 1/2 that is parallel to the z-axis. The cylinder is centered at x = 1/2on the x-axis.(c) sphere centered at the point (0.5, 0.0) with radius of
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