1. 9.8 Line integrals Integration of a function defined over an interval [a,b] Integration of a function defined along a curve C We will study Curve integral

2. Example 1: Evaluation of the Line Integral Evaluate along the the quarter-circle C (1,0) (0,1) Basic Idea is to Convert the line integral to a difinite integral Example: Step 1 Describe the curve by a vector funcion Step 2 Convert the integral into integral in terms of t only

3. Example 1: Evaluation of the Line Integral Evaluate along the the quarter-circle C (4,0) (0,4) Steps 1)Find the parametric equations of C : 2) let 3) Replace

4. In Problem 5 and 6, evaluate On the indicated curve C HW Exercises 9.8

5. Notation. or

6. In Problems 7-10, evaluate On the given curve C between (-1,2) and (2,5) Exercises 9.8 (-1,2) (2,2) (2,5) 9)10) HW (-1,2) (2,0) (2,5) (-1,0)

7. Line Integrals in the Plane Method of Evaluation Curve Defined Parametrically 1 2 3 differential of arc length

8. Line Integrals in the Plane Method of Evaluation Curve Defined Parametrically 3 differential of arc length (4,0) (0,4) Evaluate along the the quarter-circle C Example The line integral along C with respect to (wrt) arclength

9. Line Integrals in the Plane Method of Evaluation 1 2 3 differential of arc length Curve Defined by an Explicit Function

10. Curve defined by an explicit function Example 2/ pp489 Evaluate where C is given by

11. Line Integrals in Space 1 2 4 differential of arc length 3

12. In Problem 5 and 6, evaluate On the indicated curve C HW Line Integrals in Space 4 differential of arc length

13. C is a curve Terminology C is SMOOTH C is piecewise smooth

14. Terminology C is closed curve C simple closed curve Does not cross itself C is a curve A line integral along a closed curve C is very often denoted by

15. Evaluate On the closed curve C shown in figure Example 4: Closed Curve Example

16. WORK PHY-101 The work done in moving an object from x=a to x=b by a force F(x), which varies in magnitude but not in direction, is given by Math-301 F(x,y) is a force field given by F(x,y)= p(x,y) i + Q(x,y) j C is a cuve traced by r(t) = f(t) i + g(t) j WORK done by F along C or