1. Applications Of The Definite Integral The Area under the curve of a function The area between two curves

2. Example 1:1. let f (x)=1-x. Find the area bounded by the curve of f, the x-axis and the lines x=a and x=b for each of the following cases: (1) a=-1,b=1 (2) a=1,b=2 (3) a=-1,b=2 The graph: Is a straight line y=1-x: f(x) is positive on the interval [-1, 1) f(x) is negative on the interval (1, 2]

3. Solution: The area A 1 between f, the x-axis and the lines x=-1 and x=1 is: A1=A1= Case (1) Find the area bounded by the curve of f, the x-axis and the lines x=-1 and x=1.

4. Case(2): Find the area bounded by the curve of f, the x-axis and the lines x=1 and x=2 Solution: The area A 2 between f, the x-axis and the lines x=1 and x=2 is: A2=A2=

5. Solution: The area A between f,the X-axis and the lines X=-1 and X=2 is : Case(3): Find the area bounded by the curve of f, the x-axis and the lines x=-1 and x=2

6. Example (2) Let: Find the area of the region bounded by f and g from x=a to x=b for each of the following case: (1) a=-1, b=0 (2) a=0, b=1 (3) a=-1, b=1

7. Case(1): g (X)>f (X) on (-1,0) and hence on this interval, we have : g (X) –f (X)>0 So|g (X) –f (X)| =g (X)-f (X) The area A 1 between f and g from X= -1 and x=0 is :

8. Case (2): f(x) >g (X) on(0,1) and hence on this interval, we have F(X) –g (X)>0 so |g (X) –f (X)| =f (X) –g (X) = The area A 2 between f and g from X =0 to X=1 is:

9. Case (3) The area A between f and g from x = -1 to x=1