Arc Length and Surface Area Revisited
We won't be working any instances in this part. This section is here exclusively for the aim of summarizing up all the arc length and surface area problems. The arc length and surface area has arisen several times and each time we got a new formula out of the mix. Students frequently get a little overwhelmed along with all the formulas. Though, there really aren't as several formulas as it might seem at 1st glance. There is precisely one arc length formula and exactly two surface area formulas. These are as follow:
L = ∫ ds
S = ∫ 2Π y ds rotation about x - axis
S = ∫ 2Π x ds rotation about y - axis
The problems come up as we have quite a few ds's that we can utilize. Once again students frequently have trouble deciding which one to use. The instances/problems generally suggest the correct one to use. Now here is a total listing of all the ds's that we've seen and when they are employed.
If y =f (x), a < x < b then
ds = √ (1 + (dy/dx)2) dx
If x =h(y), c < y < d then
ds = √ (1 + (dx/dy)2) dy
If x =f (t), y = g (t), α < t < β then
ds = √ ((dx/dt)2 + (dy/dt)2) dt
If r = f (θ), α < θ < β then
ds = √ (r2 + (dr/dθ)2) dθ
Depending upon the type of the function we can speedily tell which ds to use.
There is just only one other thing to worry about in terms of the surface area formula.The ds will make sure a new differential to the integral. Before integrating ensure all the variables are in terms of this new differential.For instance if we have parametric equations we'll make use of the third ds and then we'll need to ensure and substitute for the x or y depending upon which axis we rotate regarding to obtain everything in terms of t.
Similarly, if we have a function in the form like x = h(y) then we'll make use of the second ds and if the rotation is regarding the y-axis we'll require to substitute for the x in the integral.Conversely if we rotate about the x-axis we won't require to do a substitution for the y.