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It follows that cos é sind, Therefore, the cone makes an angle of 45 degrees with respect to the z-axis, as shown in the following plot along with the top half of the sphere: In[485]: Clearip] plotcone = ParametricPlot3DICe Coste] Sin [Pi / 40, pinie Sin [Pi / 40, e Cos [Pi/41}, (e, 0, 2 Pi}, {p, plotsphere = ParametricPlot3D1 Coste] Sinio], Cos 161 Sin [0] Sin 161, Cos [@] Cos[01}, (e, 0, 2 Pi}, (6, o, Pi/4}1; Show [plotcone, plotsphere, PlotRange - All, ViewPoint -> (1, 1, 1/4}, ImageSize (2503) 10 0.5 -0.5 2Q's 00 ** 0.5 0.5 It is now clear that the solid Wis described by 2n, 0sd sx/4, and0spscosd. Thus, its volurne is given by the triple integral 666 which in Mathematica evaluates to In tegrate 2* Sin[0], (e, or 2 Pi}, (6, o, Pi /4}, (p, 0, Cos [61}] 71 8 Exercises In Excrcises through 4, evaluate the given double integral by converting to polar coordinates: 1. 2 3. where D is the annulus (donut-shaped region) with inner radius and outer radius 3. 4. dA, where D is the region inside the cardioid 1+ cos 1. 5. Use polar coordinates to calculate the volume of the solid that lies below the paraboloid = X \2 and inside the cylinder X /2 6. Evaluate the triple integral 6.0 vided dx by converting to cylindrical coordinates 7. Use cylindrical coordinates to calculate the triple integral 556(2+y2)dv. where W is the solid bounded between the two paraboloids x /7 and =8-7-2 8. Evaluate the triple integral d-p 4-x2-y2 (2+++2) d:dydx by converting to spherical coordinates 9. The solid defined by the spherical equation p = sin & iscalled the torus. a. Plot the torus. b. Calculate the volume of the torus. 10. Icc-Crearn Cone: A solid W in the shape of an icc-cream cone is bounded below by the cylinder < V and above by the sphere 2 +y + 2 = S. Plot W and determine its volurne.

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