7.4 Arc Length and Surface of Revolution (Photo not taken by Vickie Kelly) Greg Kelly, Hanford High School, Richland, Washington

Objectives Find the arc length of a smooth curve. Find the area of a surface of revolution.

Rectifiable curve: One that has a finite arc length f is rectifiable on [a,b] if f ' is continuous on [a,b]. If rectifiable, f is continuously differentiable on [a,b] and its graph is a smooth curve.

Lengths of Curves: If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the pythagorean theorem: Length of Curve (Cartesian) We need to get dx out from under the radical.

Length of Curve (Cartesian) (function of x) Length of Curve (Cartesian) (function of y)

Example: Find the arc length of:

Example Find the arc length of:

Example: Find the arc length of: Solve for x: When x=0: When x=8:

Example: Find the arc length of:

r Surface Area: Consider a curve rotated about the x-axis: The surface area of this band is: r The radius is the y-value of the function, so the whole area is given by: This is the same ds that we had in the “length of curve” formula, so the formula becomes: Surface Area about x-axis (Cartesian): To rotate about the y-axis, just reverse x and y in the formula!

If revolving f(x) about x-axis or g(y) about the y-axis: r(x)=f(x) r(y)=f(y)

Example: Find the area of the surface formed by revolving on [0,1] about the x-axis. r=y

If revolving f(x) about y-axis or g(y) about the x-axis: r(x)=x r(y)=y

Example: Find the area of the surface formed by revolving on about the y-axis. r=x

(Use the calculator to evaluate integrals.) Homework 7.4 (page 485) #3 – 13 odd, 17 – 25 odd (Don't graph), 37, 39, 43 (Use the calculator to evaluate integrals.) p