volume of part of a sphere?

Question

Use calculus to find the volume of a cap of a sphere with height "h" and radius "r".

HERE IS THE IMAGE!
http://hw.math.ucsb.edu/webwork/math3b-01-MB07/tmp/gif/5-prob6-prob6_2_49.gif

10 pts. for the correct reasonable answer!

Answer 1

Calculate the volume of the spherical cap.

Let
r = radius of sphere
h = height of cap

x² = r² - y²

Integrate from y = r-h to y = r

Volume = ∫πx² dy = ∫π(r² - y²) dy

= πr²y - πy³/3 | [Evaluated from r-h to r]

= (πr³ - πr³/3) - [πr²(r - h) - π(r - h)³/3]

= 2πr³/3 - πr³ + πr²h + (π/3)(r³ - 3r²h + 3rh² - h³)

= -πr³/3 + πr²h + (πr³/3 - πr²h + πrh² - πh³/3)

= (πh²/3)(3r - h)

Answer 2

Volume Of Spherical Cap

Answer 3

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RE:
volume of part of a sphere?
Use calculus to find the volume of a cap of a sphere with height "h" and radius "r".

HERE IS THE IMAGE!
http://hw.math.ucsb.edu/webwork/math3b-01-MB07/tmp/gif/5-prob6-prob6_2_49.gif

10 pts. for the correct reasonable answer!

Answer 4

well i would say the question isn't clear..it would be better if you type the question with the same wording volume of the sphere = [4/3] * pi * r^3 = 4/3 * pi * 4.5^3 calculate

Answer 5

You need spherical coordinates (the preceding answer is of course totally false).

Let F defined on [0,r]x[0,2\pi]x[0,\pi] by
F(R, S, T)=(R\cos(S)\sin(T), R\sin(S)\sin(T), R\cos(T))
(the change into spherical coordinates).
The absolute value of its Jacobian is R^2\sin(T).

Your domain of integration (cap) is defined by
{(x, y, z) : x^2 + y^2 + z^2 \leq r^2 , r-h \leq z \leq r}
and its pre-image through F is
D=[0,r]x[0,2\pi]x[0,T_0],
where T_0 satisfies \cos(T_0)=(r-h)/r=1-h/r.

Volume = \int_{D}R^2\sin(T)dRdSdT
= (\int_0^rR^2dR).(\int_0^{2\pi}dS).(\int_0^{T_0}\sin(T)dT)
= 1/3r^3.2\pi.(1-\cos(T_0)) = 2\pi.h.r^2/3.
QED.
If you need details just post another question.