1. Area of a Region Between 2 Curves
Section 6.1

2. General Solution
When determining the area between a function and the x-axis
Graph the function first
Note the zeros of the function
Split the function into portions where f(x) > 0 and f(x) < 0
Where f(x) < 0, take absolute value of the definite integral

3. Try This!
Find the area between the function h(x)=x2 – x – 6 and the x-axis
Note that we are not given the limits of integration
We must determine zeros to find limits
Also must take absolute value of the integral since specified interval has f(x) < 0

4. Area Between Two Curves
Consider the region between f(x) = x2 – 4 and g(x) = 8 – 2x2
Must graph to determine limits
Now consider function inside integral
Height of a slice is g(x) – f(x)
So the integral is

5. g f

6. Area Between Curves
Find the area of the shaded region:

7. Area Between Curves
Find the area of the shaded region:

8. Area Between Curves
In general:

9. Area Between Curves
In general:

10. 1. Find the area of the shaded region.

11. 1. Find the area of the shaded region.
Find intersection points first.

12. 2. Sketch the region represented by

13. 2. Sketch the region represented by
top
bottom

14. Representative rectangles are sometimes used
Representative rectangles are sometimes used. A vertical rectangle (of width ) implies integration with respect to x, whereas a horizontal rectangle (of width ) implies integration with respect to y.

15. If 2 curves intersect at more that 2 points, then to find the area of the region between the curves, you must find all points of intersection and check to see which curve is above the other in each interval determined by these point.

16. In general, to determine the area between 2 curves, you can use
in variable x -vertical rectangles
in variable y - horizontal rectangles

17. Area Between Two Curves
Sketch
Determine which curve is on top
Determine a and b
Integrate!

18. Example: Find the area trapped between y = x 2 and y = 1. Top curve:
Bottom curve:
y = 1
y = x 2
Solve for a and b:
x2 = 1, so x = 1 and x = –1.
The correct integral then is
Evaluate:

19. Example: Set up an integral to find the area trapped between y = x and
in the region for which x is between 2 and 5.
Can we do this exactly?
No! (Not yet.) This is not a quadratic…
this is not a first or second degree polynomial.
We know that Simpson’s rule is usually more accurate

20. Example 6 Determine the area of the region enclosed by and .

21. The area is,
                                                                                                                              

22. 1. Find the area of the region bounded by
No calculator.

23. 1. Find the area of the region bounded by
No calculator.

24. 2. Use your calculator to find the area between

25. 2. Use your calculator to find the area between
Intersection points: C B
Store these values as the respective letter!

26. 1. Find the area of the region bounded by
No calculator.

27. 1. Find the area of the region bounded by
No calculator.

28. 2. Find the area of the triangle with vertices A(2,-3),
2. Find the area of the triangle with vertices A(2,-3), B(4,6), and C(6,1).

29. 2. Find the area of the triangle with vertices A(2,-3),
2. Find the area of the triangle with vertices A(2,-3), B(4,6), and C(6,1).

30. Integration as an Accumulation Process
We can think of this as a function of b
This gives us the accumulated area under the curve on the interval [0, b]

31. Try It Out Find the accumulation function for Evaluate F(0) = 0

32. AP QUESTIONS