Posted by Saksham on Tuesday, March 12, 2013 at 9:02am.
Let V be the volume of the 3-dimensional structure bounded by the paraboloid z=1−x^2−y^2, planes x=0, y=0 and z=0 and by the cylinder x^2+y^2−x=0. If V=aπ/b, where a and b are coprime positive integers, what is the value of a+b?
calculus - Steve, Tuesday, March 12, 2013 at 2:29pm
The cylinder is centered at (1/2,0) and has radius 1/2
Seems the best way to handle this one is with cylindrical coordinates (like polar coordinates, plus a z-axis):
As with polar coordinates, the alement of area is
dA = r dr dθ
so, the volume element is
dV = r dr dθ dz
We have z = 1-(x^2+y^2) = 1-r^2
x^2+y^2-x = 0 becomes
r^2 - rcosθ = 0, or
r = 1-cosθ
V = ∫[0,pi/2] ∫[0,cosθ] z r dr dθ
= ∫[0,pi/2] ∫[0,cosθ] r(1-r^2) dr dθ
= ∫[0,pi/2] (1/2 cos^2(θ) - 1/3 cos^3(θ)) dθ
= 1/72 (18θ - 18sinθ + 9sin2θ - 2sin3θ) [0,pi/2]
= pi/8 - 2/9
Hmm. I don't get a*pi/b
Must have messed up. Check it out.
calculus - joshua, Sunday, March 17, 2013 at 12:34pm
Answer This Question
More Related Questions
- calculus - Let V be the volume of the three-dimensional structure bounded by the...
- Calculus - Let V be the volume of the three-dimensional structure bounded by the...
- calculus (point me in the right direction please?) - Let V be the volume of the ...
- mathematics - x,y and z are complex numbers that satisfy the equations x+y+z=2, ...
- math calculus - The volume of the solid obtained by rotating the region bounded ...
- biology - Given the part of the molecule shown below, can the underlined atom ...
- Calculus - Let R be the region bounded by y=−3(x−1)(x−3) and ...
- math - The number 27+200−−−√−−−−...
- Trigonometry - Let è be the angle between the x-axis and the line connecting the...
- Grade 12 Calculus - Evaluate each of the following. Show all your calculations. ...