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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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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
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