Arc Length and Surface Area

1. Arc Length and Surface Area Lesson 7.4

2. 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

3. Arc Length • We sum the individual lengths • When we take a limit of the above, we get the integral Note the New Spreadsheet Assignment

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

5. Surface Area • Suppose we rotate thef(x) from slide 2 aroundthe x-axis • A surface is formed • Think of the surface as a series of circles • What is circumference at xi? P1 Pi Pn • P0 • • • • • • xi b a

6. Surface Area • We add the circumferences of all the circles • From a to b • Over entire length of the curve

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

8. Surface Area • Surface area =

9. 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

10. Assignment • Lesson7.4 • Page 282 • Exercises 1 – 13 odd, 14