1a)Set up the integral for the volume of the solid lies within the sphere x^2+y^

Question

1a)Set up the integral for the volume of the solid lies within the sphere x^2+y^2+z^2=4 above the xy-plane and outside the cone z=sqrt(x^2+y^2).

b)Using the coordinate system indicated below, set up an integral to find the volume of the right circular cylinder of radius 1 and height 2. (Hint: The rectangular equation is x^2+y^2=1 and z is between 0 and 2.)

i)Cartesian(rectangular)
ii)Cylindrical
iii)Spherical

c) Using your work below, evaluate whicever integral you choose to find the volume of the right circular cylinder of radius 1 and height 2.

Explanation / Answer

The bounds for z are given by -?(4 - x^2 - y^2) ? z ? ?(4 - x^2 - y^2)
==> -?(4 - r^2) ? z ? ?(4 - r^2).

The projection onto the xy-plane is the disk x^2 + y^2 ? 1
==> r is in [0, 1] and ? = [0, 2?].

Thus, the volume equals
??? 1 dV
= ?(? = 0 to 2?) ?(r = 0 to 1) ?(z = -?(4 - r^2) to ?(4 - r^2)) 1 (r dz dr d?)
= 2? ?(r = 0 to 1) 2r ?(4 - r^2) dr
= 2? * (-2/3)?(4 - r^2)^(3/2) {for r = 0 to 1}
= (4?/3) [8 - 3^(3/2)].

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.