PracticeExam4.mw

> with(plots); -1; with(Student[MultivariateCalculus]); -1
 

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>
 

Exercise 0 

> MultiInt(r, r = 2 .. 3, theta = 0 .. `+`(`*`(2, `*`(Pi))), output = steps)
 

 

 

 

 

Int(Int(r, r = 2 .. 3), theta = 0 .. `+`(`*`(2, `*`(Pi))))
` ` = Int(Eval(`+`(`*`(`/`(1, 2), `*`(`^`(r, 2)))), r = 2 .. 3), theta = 0 .. `+`(`*`(2, `*`(Pi))))
` ` = Int(`/`(5, 2), theta = 0 .. `+`(`*`(2, `*`(Pi))))
` ` = Eval(`+`(`*`(`/`(5, 2), `*`(theta))), theta = 0 .. `+`(`*`(2, `*`(Pi))))
`+`(`*`(5, `*`(Pi))) (1)
 

Exercise 2 

part a 

> MultiInt(`+`(`*`(2, `*`(x, `*`(y)))), x = `+`(1, `-`(`*`(`/`(1, 3), `*`(y)))) .. `+`(3, `-`(y)), y = 0 .. 3, output = steps)
 

 

 

 

 

Int(Int(`+`(`*`(2, `*`(x, `*`(y)))), x = `+`(1, `-`(`*`(`/`(1, 3), `*`(y)))) .. `+`(3, `-`(y))), y = 0 .. 3)
` ` = Int(Eval(`*`(`^`(x, 2), `*`(y)), x = `+`(1, `-`(`*`(`/`(1, 3), `*`(y)))) .. `+`(3, `-`(y))), y = 0 .. 3)
` ` = Int(`*`(y, `*`(`+`(`*`(`^`(`+`(3, `-`(y)), 2)), `-`(`*`(`^`(`+`(1, `-`(`*`(`/`(1, 3), `*`(y)))), 2)))))), y = 0 .. 3)
` ` = Eval(`+`(`*`(`/`(2, 9), `*`(`^`(y, 4))), `-`(`*`(`/`(16, 9), `*`(`^`(y, 3)))), `*`(4, `*`(`^`(y, 2)))), y = 0 .. 3)
6 (2)
 

2b) 

>
 

> `:=`(My, MultiInt(`*`(`*`(`+`(`*`(2, `*`(x))), x), `*`(y)), x = `+`(1, `-`(`*`(`/`(1, 3), `*`(y)))) .. `+`(3, `-`(y)), y = 0 .. 3))
 

`/`(39, 5) (3)
 

> `:=`(Mx, MultiInt(`*`(`+`(`*`(2, `*`(x, `*`(y)))), y), x = `+`(1, `-`(`*`(`/`(1, 3), `*`(y)))) .. `+`(3, `-`(y)), y = 0 .. 3))
 

`/`(36, 5) (4)
 

> `:=`(COM, [`+`(`*`(`/`(1, 6), `*`(My))), `+`(`*`(`/`(1, 6), `*`(Mx)))])
 

[`/`(13, 10), `/`(6, 5)] (5)
 

>
 

2c) 

> `:=`(M2x, MultiInt(`*`(`+`(`*`(2, `*`(`^`(y, 2), `*`(x)))), y), x = `+`(1, `-`(`*`(`/`(1, 3), `*`(y)))) .. `+`(3, `-`(y)), y = 0 .. 3))
 

`/`(54, 5) (6)
 

Exercise 4 

> MultiInt(sqrt(`+`(2, `*`(`^`(x, 2)))), x = 0 .. 1, y = 0 .. 1); 1; evalf(%)
 

 

`+`(`*`(`/`(1, 2), `*`(`^`(3, `/`(1, 2)))), arcsinh(`+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2)))))))
1.524504352 (7)
 

>
 

Exercise 5 

> MultiInt(`*`(sqrt(`+`(1, `*`(4, `*`(`^`(r, 2))))), `*`(r)), r = 0 .. 1, theta = 0 .. `+`(`*`(2, `*`(Pi))), output = steps); 1; evalf(%)
 

 

 

 

 

 

Int(Int(`*`(`^`(`+`(1, `*`(4, `*`(`^`(r, 2)))), `/`(1, 2)), `*`(r)), r = 0 .. 1), theta = 0 .. `+`(`*`(2, `*`(Pi))))
` ` = Int(Eval(`+`(`*`(`/`(1, 12), `*`(`^`(`+`(1, `*`(4, `*`(`^`(r, 2)))), `/`(3, 2))))), r = 0 .. 1), theta = 0 .. `+`(`*`(2, `*`(Pi))))
` ` = Int(`+`(`-`(`/`(`*`(`/`(1, 8), `*`(`+`(`-`(`*`(`/`(10, 3), `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`(1, 2)))))), `*`(`/`(2, 3), `*`(`^`(Pi, `/`(1, 2))))))), `*`(`^`(Pi, `/`(1, 2)))))), theta = 0 .....
` ` = Eval(`+`(`-`(`/`(`*`(`/`(1, 8), `*`(`+`(`-`(`*`(`/`(10, 3), `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`(1, 2)))))), `*`(`/`(2, 3), `*`(`^`(Pi, `/`(1, 2))))), `*`(theta))), `*`(`^`(Pi, `/`(1, 2)))))),...
`+`(`-`(`*`(`/`(1, 4), `*`(`^`(Pi, `/`(1, 2)), `*`(`+`(`-`(`*`(`/`(10, 3), `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`(1, 2)))))), `*`(`/`(2, 3), `*`(`^`(Pi, `/`(1, 2))))))))))
5.330413500 (8)
 

 

Exercise 6 

 

a)  Volume 

 

> MultiInt(1, z = 0 .. `+`(4, `-`(`*`(2, `*`(y)))), y = 0 .. sqrt(`+`(4, `-`(`*`(`^`(x, 2))))), x = 0 .. 2, output = steps)
 

 

 

 

 

 

 

Int(Int(Int(1, z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(Eval(z, z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(`+`(4, `-`(`*`(2, `*`(y)))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Eval(`+`(`*`(4, `*`(y)), `-`(`*`(`^`(y, 2)))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(`+`(`*`(4, `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), `-`(4), `*`(`^`(x, 2))), x = 0 .. 2)
` ` = Eval(`+`(`*`(2, `*`(x, `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2))))), `*`(8, `*`(arcsin(`+`(`*`(`/`(1, 2), `*`(x)))))), `-`(`*`(4, `*`(x))), `*`(`/`(1, 3), `*`(`^`(x, 3)))), x = 0 .. 2)
`+`(`*`(4, `*`(Pi)), `-`(`/`(16, 3))) (9)
 

 

b)  Mass 

 

> `:=`(Mass, MultiInt(x, z = 0 .. `+`(4, `-`(`*`(2, `*`(y)))), y = 0 .. sqrt(`+`(4, `-`(`*`(`^`(x, 2))))), x = 0 .. 2, output = steps))
 

 

 

 

 

 

 

Int(Int(Int(x, z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(Eval(`*`(x, `*`(z)), z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(`*`(x, `*`(`+`(4, `-`(`*`(2, `*`(y)))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Eval(`*`(x, `*`(`+`(`*`(4, `*`(y)), `-`(`*`(`^`(y, 2)))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(`+`(`-`(`*`(x, `*`(`+`(4, `-`(`*`(`^`(x, 2))))))), `*`(4, `*`(x, `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))))), x = 0 .. 2)
` ` = Eval(`+`(`*`(`/`(1, 4), `*`(`^`(x, 4))), `-`(`*`(2, `*`(`^`(x, 2)))), `-`(`*`(`/`(4, 3), `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(3, 2)))))), x = 0 .. 2)
`/`(20, 3) (10)
 

>
 

 

Moment about y-z plane 

 

>
 

> `:=`(Myz, MultiInt(`*`(`^`(x, 2)), z = 0 .. `+`(4, `-`(`*`(2, `*`(y)))), y = 0 .. sqrt(`+`(4, `-`(`*`(`^`(x, 2))))), x = 0 .. 2, output = steps))
 

 

 

 

 

 

 

Int(Int(Int(`*`(`^`(x, 2)), z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(Eval(`*`(`^`(x, 2), `*`(z)), z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(`*`(`^`(x, 2), `*`(`+`(4, `-`(`*`(2, `*`(y)))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Eval(`*`(`^`(x, 2), `*`(`+`(`*`(4, `*`(y)), `-`(`*`(`^`(y, 2)))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(`+`(`-`(`*`(`^`(x, 2), `*`(`+`(4, `-`(`*`(`^`(x, 2))))))), `*`(4, `*`(`^`(x, 2), `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))))), x = 0 .. 2)
` ` = Eval(`+`(`*`(`/`(1, 5), `*`(`^`(x, 5))), `-`(`*`(`/`(4, 3), `*`(`^`(x, 3)))), `-`(`*`(x, `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(3, 2))))), `*`(2, `*`(x, `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/...
`+`(`-`(`/`(64, 15)), `*`(4, `*`(Pi))) (11)
 

 

Moment about the x-z plane 

 

> `:=`(Mxz, MultiInt(`*`(y, `*`(x)), z = 0 .. `+`(4, `-`(`*`(2, `*`(y)))), y = 0 .. sqrt(`+`(4, `-`(`*`(`^`(x, 2))))), x = 0 .. 2, output = steps))
 

 

 

 

 

 

 

Int(Int(Int(`*`(y, `*`(x)), z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(Eval(`*`(y, `*`(x, `*`(z))), z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(`*`(y, `*`(x, `*`(`+`(4, `-`(`*`(2, `*`(y))))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Eval(`*`(x, `*`(`+`(`-`(`*`(`/`(2, 3), `*`(`^`(y, 3)))), `*`(2, `*`(`^`(y, 2)))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(`+`(`-`(`*`(`/`(2, 3), `*`(x, `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(3, 2)))))), `*`(2, `*`(x, `*`(`+`(4, `-`(`*`(`^`(x, 2)))))))), x = 0 .. 2)
` ` = Eval(`+`(`*`(`/`(2, 15), `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(5, 2)))), `-`(`*`(`/`(1, 2), `*`(`^`(x, 4)))), `*`(4, `*`(`^`(x, 2)))), x = 0 .. 2)
`/`(56, 15) (12)
 

 

Moment about the x-y plane 

 

> `:=`(Mxy, MultiInt(`*`(z, `*`(x)), z = 0 .. `+`(4, `-`(`*`(2, `*`(y)))), y = 0 .. sqrt(`+`(4, `-`(`*`(`^`(x, 2))))), x = 0 .. 2, output = steps))
 

 

 

 

 

 

 

Int(Int(Int(`*`(z, `*`(x)), z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(Eval(`+`(`*`(`/`(1, 2), `*`(`^`(z, 2), `*`(x)))), z = 0 .. `+`(4, `-`(`*`(2, `*`(y))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Int(`+`(`*`(`/`(1, 2), `*`(x, `*`(`^`(`+`(4, `-`(`*`(2, `*`(y)))), 2))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(Eval(`+`(`-`(`*`(`/`(1, 12), `*`(x, `*`(`^`(`+`(4, `-`(`*`(2, `*`(y)))), 3)))))), y = 0 .. `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1, 2)))), x = 0 .. 2)
` ` = Int(`+`(`*`(`/`(2, 3), `*`(x, `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(3, 2))))), `-`(`*`(4, `*`(x, `*`(`+`(4, `-`(`*`(`^`(x, 2)))))))), `*`(8, `*`(x, `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(1,...
` ` = Eval(`+`(`-`(`*`(`/`(2, 15), `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(5, 2))))), `*`(`^`(x, 4)), `-`(`*`(8, `*`(`^`(x, 2)))), `-`(`*`(`/`(8, 3), `*`(`^`(`+`(4, `-`(`*`(`^`(x, 2)))), `/`(3, 2))))...
`/`(48, 5) (13)
 

 

Center of Mass 

> [`/`(`*`(Myz), `*`(Mass)), `/`(`*`(Mxz), `*`(Mass)), `/`(`*`(Myz), `*`(Mass))]
 

[`+`(`-`(`/`(16, 25)), `*`(`/`(3, 5), `*`(Pi))), `/`(14, 25), `+`(`-`(`/`(16, 25)), `*`(`/`(3, 5), `*`(Pi)))] (14)
 

 

 

> MultiInt(`*`(`^`(r, 2), `*`(cos(theta))), z = 0 .. `+`(4, `-`(`*`(2, `*`(r, `*`(sin(theta)))))), r = 0 .. 2, theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))), output = steps)
 

 

 

 

 

 

 

Int(Int(Int(`*`(`^`(r, 2), `*`(cos(theta))), z = 0 .. `+`(4, `-`(`*`(2, `*`(r, `*`(sin(theta))))))), r = 0 .. 2), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Int(Int(Eval(`*`(`^`(r, 2), `*`(cos(theta), `*`(z))), z = 0 .. `+`(4, `-`(`*`(2, `*`(r, `*`(sin(theta))))))), r = 0 .. 2), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Int(Int(`*`(`^`(r, 2), `*`(cos(theta), `*`(`+`(4, `-`(`*`(2, `*`(r, `*`(sin(theta))))))))), r = 0 .. 2), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Int(Eval(`*`(cos(theta), `*`(`+`(`-`(`*`(`/`(1, 2), `*`(sin(theta), `*`(`^`(r, 4))))), `*`(`/`(4, 3), `*`(`^`(r, 3)))))), r = 0 .. 2), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Int(`+`(`-`(`*`(8, `*`(cos(theta), `*`(sin(theta))))), `*`(`/`(32, 3), `*`(cos(theta)))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Eval(`+`(`*`(4, `*`(`^`(cos(theta), 2))), `*`(`/`(32, 3), `*`(sin(theta)))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
`/`(20, 3) (15)
 

 

Exercise 7: 

 

> `:=`(TheCyl, implicitplot3d(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2))) = `+`(`*`(2, `*`(y))), x = -2 .. 2, y = -2 .. 2, z = 0 .. 4, grid = [20, 20, 20], color = red, style = patchnogrid, transparency = .5));...
`:=`(TheCyl, implicitplot3d(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2))) = `+`(`*`(2, `*`(y))), x = -2 .. 2, y = -2 .. 2, z = 0 .. 4, grid = [20, 20, 20], color = red, style = patchnogrid, transparency = .5));...
 

> `:=`(TheCone, implicitplot3d(`+`(`*`(`^`(z, 2)), `-`(`*`(`^`(x, 2))), `-`(`*`(`^`(y, 2)))) = 0, x = -2 .. 2, y = -2 .. 2, z = 0 .. 4, grid = [30, 30, 30], style = patchnogrid, transparency = .5)); -1
`:=`(TheCone, implicitplot3d(`+`(`*`(`^`(z, 2)), `-`(`*`(`^`(x, 2))), `-`(`*`(`^`(y, 2)))) = 0, x = -2 .. 2, y = -2 .. 2, z = 0 .. 4, grid = [30, 30, 30], style = patchnogrid, transparency = .5)); -1
 

> display(TheCyl, TheCone, axes = normal)
 

Plot
 

> MultiInt(r, z = 0 .. r, r = 0 .. `+`(`*`(2, `*`(sin(theta)))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))), output = steps)
 

 

 

 

 

 

 

Int(Int(Int(r, z = 0 .. r), r = 0 .. `+`(`*`(2, `*`(sin(theta))))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Int(Int(Eval(`*`(r, `*`(z)), z = 0 .. r), r = 0 .. `+`(`*`(2, `*`(sin(theta))))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Int(Int(`*`(`^`(r, 2)), r = 0 .. `+`(`*`(2, `*`(sin(theta))))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Int(Eval(`+`(`*`(`/`(1, 3), `*`(`^`(r, 3)))), r = 0 .. `+`(`*`(2, `*`(sin(theta))))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Int(`+`(`*`(`/`(8, 3), `*`(`^`(sin(theta), 3)))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
` ` = Eval(`+`(`-`(`*`(`/`(8, 9), `*`(`^`(sin(theta), 2), `*`(cos(theta))))), `-`(`*`(`/`(16, 9), `*`(cos(theta))))), theta = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))))
`/`(16, 9) (16)
 

>
 

 

Exercise 8 

 

> MultiInt(`*`(`*`(rho, `*`(`^`(rho, 2))), `*`(sin(phi))), rho = 0 .. 4, phi = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))), theta = 0 .. Pi, output = steps)
 

 

 

 

 

 

 

Int(Int(Int(`*`(`^`(rho, 3), `*`(sin(phi))), rho = 0 .. 4), phi = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi)))), theta = 0 .. Pi)
` ` = Int(Int(Eval(`+`(`*`(`/`(1, 4), `*`(`^`(rho, 4), `*`(sin(phi))))), rho = 0 .. 4), phi = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi)))), theta = 0 .. Pi)
` ` = Int(Int(`+`(`*`(64, `*`(sin(phi)))), phi = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi)))), theta = 0 .. Pi)
` ` = Int(Eval(`+`(`-`(`*`(64, `*`(cos(phi))))), phi = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi)))), theta = 0 .. Pi)
` ` = Int(64, theta = 0 .. Pi)
` ` = Eval(`+`(`*`(64, `*`(theta))), theta = 0 .. Pi)
`+`(`*`(64, `*`(Pi))) (17)
 

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