## Transcribed Text

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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