Arc Length and Surface Area Lesson 6.4

What Is Happening?

What is another way of representing this? Arc Length We seek the distance along the curve from f(a) to f(b) That is from P0 to Pn The distance formula for each pair of points P1 • Pi P0 Pn • • • • • b a Why? What is another way of representing this?

Arc Length We sum the individual lengths When we take a limit of the above, we get the integral

Arc Length Find the length of the arc of the function for 1 < x < 2

Surface Area of a Cone Slant area of a cone Slant area of frustum s h

Surface Area Suppose we rotate the f(x) from slide 2 around the x-axis A surface is formed A slice gives a cone frustum P1 Pi P0 Pn • • • • • • • xi b a Δs

Surface Area We add the cone frustum areas of all the slices From a to b Over entire length of the curve

Surface Area Consider the surface generated by the curve y2 = 4x for 0 < x < 8 about the x-axis

Surface Area Surface area =

Limitations We are limited by what functions we can integrate Integration of the above expression is not trivial We will come back to applications of arc length and surface area as new integration techniques are learned

Assignment Lesson 6.4 Page 393 Exerxises 1 – 25 odd