Section 8.1 Continued
Yesterday we created an integral that would give us the length of a curve.
Does this answer make sense? Consider the graph and get the approximate length using the Pythagorean theorem.
We can use the arc length formula to create a cumulative arc length function. This function, once created, will allow us to pick any x­value and it will produce the length of the curve up to that point. (The idea behind creating a cumulative function for length and for area is important in statistics.)
We will get a function of x.
Don't stop at b. Use the variable x instead in the integral.
It will give the length of the curve from x=a to x=?
Example:
Build the cumulative length function for the function f(t).
It's a cute little u­sub.
This function gives the length of the curve from x=0 to x= whatever x we want.
Let's use the function:
Why is this a perfectly reasonable answer?
Section 8.2: Area of a Surface
We know some surface area formulas from geometry...surface area of a sphere...and a cylinder...
Remember that surface area means we are not counting any internal stuff.
So for an open top/bottom cylinder, we can find the surface area using the expression:
Right Circular Cylinder
Cut the side of the cylinder and lay it flat.
Can we use what we learned yesterday to generate the surface area of any shaped object?
We are taking a small thickness (piece of the arc) and rotating it, so we need our arc length integral along with the surface area of a cylinder idea.
We are adding up thin cylinders, so the overarching formula that applies is the 2 r h.
Example: Let's see this in action. Consider our old buddy the parabola. We will rotate it around the y­axis.