Arc Length and Surfaces of Revolution

Arc Length and Surfaces of Revolution Lesson 7.4

What Is Happening?

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

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

s h r Surface Area of a Cone • Slant area of a cone • Slant area of frustum L

Surface Area Δx • Suppose we rotate thef(x) from slide 2 aroundthe x-axis • A surface is formed • A slice gives a cone frustum P1 Pi Pn • P0 • • • • • • 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 7.4 • Page 383 • Exercises 1 – 29 odd also 37 and 55,

Arc Length and Surfaces of Revolution