complete final soultion
Question # 1: Marks 5
If is continuous on a closed interval [0, 1], then by using Mean-Value Theorem for integrals, show that there exits at least one value in [0, 1].
Question # 3: Marks 5
Find the volume of the spheroid formed by the revolution of the area bounded by the ellipse about the major axis .
Use parametric form
x = a cos t
y = b sin t, 0 ≤ t ≤ π
which gives the half ellipse with positive y
dx/dt = - a sin t
dy/dt = b cos t
V = π∫(a cos t)²(b cos t) dt on [0,π]
thanks to the symmetry wrt y-axis we have
V = 2πa²b∫(cos t)²(cos t) dt on [0,π/2] = (*)
∫(cos t)³ dt = ∫(cos t)(1 - (sin t)²) dt =
sin t = u
cos t dt = du
when t = 0, u = 0
when t = π/2, u = 1
= 2πa²b ∫(1 - u²) du on [0,1]
a primitive is
F(u) = 2πa²b(u - (1/3) u³)
then volume is
V = F(1) - F(0) = (4/3)πa²b
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mit student Thanks for sharing ur idea
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important note for all checks attachment for complte soution,
sooooooooooooooooory alot mera sa disscussion add karta Q1 And Q2 Ka soution miss ho gya ha bt attachment ma complte ha
Is this correct?