Name ___________________________                        BC Calculus                        SHW 5


Formulas for Arc (Curve) Length in the Plane:

`Arc Length for a curve `*y = f(x)*`:   `*Int((1+`f '`(x)^2)^(1/2),x = a .. b)


`Arc Length for a curve `*x = g(y)*`:   `*Int((1+`g '`(y)^2)^(1/2),y = c .. d)


`Arc Length for a parametric curve `*[x, y] = [x(t), y(t)]*`:   `*Int((`x '`(t)^2+` y'`(t)^2)^(1/2),t = t[1] .. t[2])


Arc Length Exercises:

`Arc Length Exercise 5.1 [Use Integration by Parts]:   `*y = ln(1-x^2), x = 0 .. 1/2




`Arc Length Exercise 5.2 [Use Hyperbolic Functions]:   `*`y = cosh(x)` = (exp(x)+exp(-x))/`2`, x = 0 .. 1




`Arc Length Exercise 5.3:   `*y = exp(x), x = 0 .. 1




`Arc Length Exercise 5.4:   `*x = t/(1+t), `  `*y = ln(1+t), t = 0 .. 2




Formulas for Surface Area:

`Surface Area rotate about x-axis` = Int(2*Pi*f(x)*(1+`f '`(x)^2)^(1/2),x = a .. b)


`Surface Area rotate about y-axis` = Int(2*Pi*g(y)*(1+`g '`(y)^2)^(1/2),y = c .. d)


`Parametric Surface Area rotate about x-axis` = Int(2*Pi*y(t)*(`x '`(t)^2+` y'`(t)^2)^(1/2),t = t[1] .. t[2])


`Parametric Surface Area rotate about y-axis` = Int(2*Pi*x(t)*(`x '`(t)^2+` y'`(t)^2)^(1/2),t = t[1] .. t[2])


Surface Area Exercises:

`Surface Area Exercise 5.1:   `*y = (4-x^2)^(1/2), x = -1 .. 1, `rotate about x-axis.`

`Surface Area Exercise 5.2:   `*x = y^(1/2), ` from `*[1, 1]*` to `*[2, 4]*`, choose axis.`




`Surface Area Exercise 5.3:   `*y = exp(x), x = 0 .. 1, `rotate about x-axis.`




`Surface Area Exercise 5.4:   `*y = sin(x), x = 0 .. Pi, `rotate about x-axis.`




`Surface Area Exercise 5.5:   `*`y = cosh(x)` = (exp(x)+exp(-x))/`2`, x = 0 .. 1, `rotate about x-axis.`




`Surface Area Exercise 5.6:   r is constant and,`*x = r*cos(t), `  `*y = r*sin(t), t = 0 .. Pi, `rotate about x-axis.`




`Surface Area Exercise 5.7:   r is constant and,`*x = r*(t-sin(t)), `  `*y = r*(1-cos(t)), t = 0 .. 2*Pi, `rotate about x-axis.`




`Surface Area Exercise 5.8:   `*x = t^3, `  `*y = t^4, t = 0 .. 1, `rotate about y-axis.`