1. Section 7.1 – Area of a Region Between Two Curves

2. White Board Challenge The circle below is inscribed into a square:
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The circle below is inscribed into a square:
What is the shaded area?
20 cm

3. White Board Challenge Calculator
Find the area of the region bounded by the function below and the x-axis between x = 1 to x = 6:

4. Area Between Two Curves
The area of a region that is bounded above by one curve, y = f(x), and below by another y = g(x).
The area is always POSITIVE.

5. White Board Challenge
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Find the area of the region between y = sec2x and y = sin x from x = 0 to x = π/4:
Subtracting the bottom area from the top, leaves only the area in-between.
TOP
BOTTOM
TOP
BOTTOM
Between
Outside
In this example, all of the area was above the x-axis. Does the same process work for “negative” area?

6. Area Between Two Curves: Positive and Negative Area
Find the area of the region between the two curves from x = a to x = b:
Subtracting the negative area switches it to adding a positive version.
TOP
BOTTOM
Between
(Positive)
TOP
BOTTOM
Between
(Negative)
THE SAME!
In this example, one area was positive and one was negative. Does the same process work if both areas are negative?
Must be positive!

7. Area Between Two Curves: Negative Area Only
Find the area of the region between the two curves from x = a to x = b:
Outside
TOP
BOTTOM
(Counted Twice)
Between (Negative)
TOP
BOTTOM
THE SAME!
Subtracting the negative area switches it to adding a positive version AND cancels the outside area.
In this example, both areas were negative. Now we can apply the three scenarios to any two curves.

8. Area Between Two Curves: A Mix
Find the area of the region between the two curves from x = a to x = b:
TOP
BOTTOM
NEG-NEG
TOP
BOTTOM
POS-POS
POS-NEG

9. Area Between Two Curves
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), below by y = g(x), on the left by x = a, and on the right by x = b is:
TOP
BOTTOM

10. Reminder: Riemann Sums
Recall that the integral is a limit of Riemann Sums:

11. Example 1
Find the area of the region between the graphs of the functions
Sketch a Graph
Find the Boundaries/Intersections
Base = dx
Integrate the Area of Each Generic Rectangle
Height = f – g
Make Generic “Riemann” Rectangle(s)

12. Example 2
Find the area of the region enclosed by the parabolas y = x2 and y = 2x – x2.
Sketch a Graph
Find the Boundaries/Intersections
Base = dx
Integrate the Area of Each Generic Rectangle
Height = (2x–x2)–(x2)
Make Generic “Riemann” Rectangle(s)

13. Example 3
Find the area of the region bounded by the graphs y = 8/x2, y = 8x, and y = x.
Sketch a Graph
Find the Boundaries/Intersections
Height = 8/x2-x
Base = dx
Base = dx
Height = 8x-x
Integrate the Area of Each Generic Rectangle
Make Generic “Riemann” Rectangle(s)

14. Example 4
Find the area of the region bounded by the curves y = sin x, y = cos x, x = 0, and x = π/2.
Sketch a Graph
Find the Boundaries/Intersections
Base = dx
Base = dx
Integrate the Area of Each Generic Rectangle
Height = cos-sin
Height = sin-cos
What other Integrals could be used?
Make Generic “Riemann” Rectangle(s)
(Symmetrical)
(Keeps it Positive)

15. Area Between Two Curves
If f and g are continuous functions on the interval [a,b], then the area of the region bounded by y = f(x), y = g(x), on the left by x = a, and on the right by x = b is:
It does not matter which function is greater.
NOTE: There have been AP problems in the past that ask for an integral without an absolute value. So the first method is still important.

16. “Warm-up”: 1985 Section I
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NOW WE CAN DO!

17. White Board Challenge
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Find the area enclosed by the line y = x – 1 and the parabola y2 = 2x + 6.

18. Example 5
Find the area enclosed by the line y = x – 1 and the parabola y2 = 2x + 6.
Sketch a Graph
Sometimes Solve for x
Find the Boundaries/Intersections
Base = dy
Height=(y+1)–(1/2y2–3)
Integrate the Area of Each Generic Rectangle
Make Generic “Riemann” Rectangle(s)

19. White Board Challenge
Using two methods (one with dx and one with dy), find the area between the x-axis and the two curves: