Arc Length and Surface Area Lesson 7 4

Arc Length and Surface Area Lesson 7. 4

What Is Happening? 2

Arc Length • We seek the distance along the curve from f(a) to f(b) § That is from P 0 to Pn P P 0 • 1 • • a Pi • Pn • • b • The distance formula for each pair of points What is another way of representing this? Why? 3

Arc Length • We sum the individual lengths • When we take a limit of the above, we get the integral 4

Arc Length • Find the length of the arc of the function for 1 < x < 2 5

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

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 § Δx P P 0 • 1 • • a Pi Pn • • • xi • b 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 y 2 = 4 x for 0 < x < 8 about the x-axis 9

Surface Area • Surface area = 10

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 11

Assignment • Lesson 7. 4 • Page 282 • Exerxises 1 – 13 odd, 14 12