Math 251-copyright Joe Kahlig, 15A Page 11. True or FalseF 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=ax2dx +Zdy=cy2dyT y =RRDyρ(x, y)dARRDρ(x, y)dAFZ20Z√4−x2−√4−x2g(x, y)dydx =Z20Zπ0g(r cos(θ), r sin(θ))rdθdrT If a,b,c,d are constants, thenZbx=aZdy=cf(x, y)dydx =Zdy=cZbx=af(x, y)dxdy2.π/2Z0sin(2θ)Z03r2cos2θ + r sin θr drdθ3. 7204.πZ03Z027−3r2Z−√9−r21rdzdrdθ5.9Z0√9−zZ0(3−y)/3Z0f(x, y, z)dxdydz6. Reverse the order of integration and then integrate. answer29283/2− 17. (a)π/2Z02Z0√16−r2Zr√3rzpr2+ z2dzdrdtheta(b) note: the sperical coordinates for the cone z2= 3x2+ 3y2is φ = π/6π/2Z0π/6Z04Z0ρ4cos φ sin φdρdφdθ8.1Z02−2xZ010−10x−5yZ01dzdydxMath 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 φdρdφdθ10.2πZ05Z2r cos θ+25Z0r2sin θdzdrdtheta11. The density at any point is proportional to the distance from the point (1, 1)ρ(x, y) = kp(x − 3)2+ (y − 1)2√2Z−√24−y2Zy2kp(x − 3)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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