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Assignment no,2

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 .

Answer

 

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 =
substitute 
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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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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