1. Area Between Two Curves
Objective: To find the area between two curves.

2. Riemann Sums
Let’s review what a Riemann Sum is:

3. Area Between Two Curves
First Area Problem: Suppose that f and g are continuous functions on an interval [a, b] and f(x) > g(x) for a < x < b.
This means that the curve y = f(x) lies above the curve y = g(x) and that the two can touch but never cross.
Find the area A of the region bounded above by y = f(x), below by y = g(x), and on the sides by the lines x = a and x = b.

4. Area Between Two Curves
To solve this problem, we divide the interval [a, b] into n subintervals, which has the effect of subdividing the region into n strips. If we assume that the width of the kth strip is , then the area of the strip can be approximated by the area of a rectangle of width and height , where is a point in the kth subinterval.

5. Area Between Two Curves
Adding these approximations yields the following Riemann Sum that approximates the area A:

6. Area Between Two Curves
Adding these approximations yields the following Riemann Sum that approximates the area A:
Taking the limit as n increases and the widths of the subintervals approach zero yields the following definite integral for the area A between the curves:

7. Area Between Two Curves
Area Formula
If f and g are continuous functions on the interval [a, b], and if f(x) > g(x) for all x in [a, b], then the area of the region bounded above by y = f(x) and below by y = g(x), on the left by the line x = a, and on the right by the line x = b is

8. Example 1
Find the area of the region bounded above by y = x + 6, bounded below by y = x2, and bounded on the sides by the lines x = 0 and x = 2.

9. Example 1
Find the area of the region bounded above by y = x + 6, bounded below by y = x2, and bounded on the sides by the lines x = 0 and x = 2.

10. Example 2
Find the area of the region that is enclosed between the curves y = x2 and y = x + 6.

11. Example 2
Find the area of the region that is enclosed between the curves y = x2 and y = x + 6.
Looking at the graph, we see
that y = x2 is the lower bound
and y = x + 6 is the upper bound.
We need to find the points of
intersection to find a and b. We
will do this with our calculator.

12. Example 2
Find the area of the region that is enclosed between the curves y = x2 and y = x + 6.
Now, we integrate to find
the answer. N

13. Area
In the case where both f and g are nonnegative on the interval [a, b], the area A between the curves can be obtained by subtracting the area under y = g(x) from the area under y = f(x).

14. Example 4
Find the area of the region enclosed by x = y2 and y = x – 2.

15. Example 4
Find the area of the region enclosed by x = y2 and y = x – 2.
The situation that makes this
problem different is that the
bottom curve is not the same
everywhere. We need to look
at this as two separate areas
and integrate twice.

16. Example 4
Find the area of the region enclosed by x = y2 and y = x – 2.
The top curve is always
, but the bottom curve
is from 0-1, and it is
y = x – 2 from The two
integrals will be:

17. Example 4
Find the area of the region enclosed by x = y2 and y = x – 2.

18. Reversing the Rolls of x and y
Sometimes it is possible to avoid splitting a region into parts by integrating with respect to y rather than x. We will now look at this situation.

19. Second Area Problem
Suppose that w and v are continuous functions of y on an interval [c, d] and that
for
This means that
lies to the right of
and that the two curves
can touch but never cross.

20. Area Formula 7.1.4 If w and v are continuous functions and if
for all y in [c, d], then the area of the region bounded on the left by , on the right by
below by y = c, and above by y = d is

21. Example 5
Find the area of the region enclosed by x = y2 and y = x – 2, integrating with respect to y.

22. Example 5
Find the area of the region enclosed by x = y2 and y = x – 2, integrating with respect to y.
First, we need to solve each
equation for x to put it in terms
of y. We also need to find the
bounds in terms of y.
x = y2
x = y + 2
c = -1, d = 2

23. Example 5
Find the area of the region enclosed by x = y2 and y = x – 2, integrating with respect to y.
This leads us to the integral:

24. Under vs. Between
When we found the area under a curve, we were only dealing with one curve and it was possible to have what we called “negative area”. Now, with the area between two curves, we will always have two curves and the area will always be positive.

25. Under vs. Between
When we found the area under a curve, we were only dealing with one curve and it was possible to have what we called “negative area”. Now, with the area between two curves, we will always have two curves and the area will always be positive.
Sometimes, the second curve will be the x or y-axis. It may be the top curve or the bottom curve. For some of these, we will need to use our knowledge of piecewise functions to solve.

26. Example 6
Find the area between the curves and y = 0 from

27. Example 6 Find the area between the curves and y = 0 from .
From , the top curve is y = sinx and the bottom curve is y = 0.
From , the top curve is y = 0 and the bottom curve is y = sinx.

28. Example 6
Find the area between the curves and y = 0 from

29. Homework
Section 6.1
1-19 odd 35

30. #7
Find the area between the curves.

31. #9
Find the area between the curves.

32. #11
Find the area between the curves.

33. #13
Find the area between the curves.

34. #15
Find the area between the curves.

35. #15 Find the area between the curves. From -1 to 0, the bottom curve
is y = -x.
From 0 to 1, the bottom curve
is y = x.
The top curve is always the
same.

36. #15 Find the area between the curves. From -1 to 0, the bottom curve
is y = -x.
From 0 to 1, the bottom curve
is y = x.
The top curve is always the
same.

37. #15 Find the area between the curves. From -1 to 0, the bottom curve
is y = -x.
From 0 to 1, the bottom curve
is y = x.
The top curve is always the
same.

38. #17
Find the area between the curves.

39. #17 Find the area between the curves. From -5 to 1, the bottom
curve is 2 – (x – 1).
From 1 to 5, the bottom
curve is 2 + (x – 1).
The top curve is always the
same.

40. #17 Find the area between the curves. From -3.33 to 1, the bottom
curve is 2 – (x – 1).
From 1 to 7.5, the bottom
curve is 2 + (x – 1).
The top curve is always the
same.

41. #17 Find the area between the curves. From -3.33 to 1, the bottom
curve is 2 – (x – 1).
From 1 to 7.5, the bottom
curve is 2 + (x – 1).
The top curve is always the
same.

42. #19
Find the area between the curves.

43. #19 Find the area between the curves. From 0 to 1, the top curve is
and the bottom
curve y = 0.
From 1 to 3, the top curve is
y = 0 and the bottom curve is

44. #19 Find the area between the curves. From 0 to 1, the top curve is
and the bottom
curve y = 0.
From 1 to 3, the top curve is
y = 0 and the bottom curve is

45. #19 Find the area between the curves. From 0 to 1, the top curve is
and the bottom
curve y = 0.
From 1 to 3, the top curve is
y = 0 and the bottom curve is