Question
Download Solution PDFThe volume of the solid surrounded by the surface
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Cartesian system (a, b, c) to spherical coordinate system (r, θ, ϕ) conversion.
a = r sin θ cos ϕ, b = r sin θ sin ϕ
c = r cos θ [note: a, b, c are generalised cartesian parameter different then given solid parameter]
da ⋅ db ⋅ dc = r2sin θ dr ⋅ dθ ⋅ dϕ
Calculation:
Given solid is
Now, put x = au3, y = bv3 & z = cω3 ---(2)
⇒ dx = 3a42⋅d4, dy = 3bv2⋅dv & dz = 3cw2dw ---(3)
Put equation (2) in equation (1), then we get
u2 + v2 + w2 = 1 (which is equation of sphere)
Since volume enclosed by solid can be calculate as,
Volume
So put equation (3) in equation (4), then we get
Volume
Now apply cartesion to spherical conversion in equation (5)
Put equation (6) into equation (5) then we get
Volume
⇒ Volume
⇒ Volume
On solving above integrals we get.
⇒ Volume
Last updated on Jul 2, 2025
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