1. The Area Between Two Curves
Lesson 6.1

2. What If … ? We want to find the area between f(x) and g(x) ?
Any ideas?

3. When f(x) < 0
Consider taking the definite integral for the function shown below.
The integral gives a negative area (!?)
We need to think of this in a different way a b
f(x)

4. Recall our look at odd functions on the interval [-a, a]
Another Problem
What about the area between the curve and the x-axis for y = x3
What do you get for the integral?
Since this makes no sense – we need another way to look at it
Recall our look at odd functions on the interval [-a, a]

5. Solution We can use one of the properties of integrals
We will integrate separately for -2 < x < 0 and 0 < x < 2
We take the absolute value for the interval which would give us a negative area.

6. 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

7. 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

8. 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

9. The Area of a Shark Fin Consider the region enclosed by
Again, we must split the region into two parts
0 < x < 1 and 1 < x < 9

10. Slicing the Shark the Other Way
We could make these graphs as functions of y
Now each slice is y by (k(y) – j(y))

11. Practice Determine the region bounded between the given curves
Find the area of the region

12. Horizontal Slices
Given these two equations, determine the area of the region bounded by the two curves
Note they are x in terms of y

13. Assignments A
Lesson 7.1A
Page 452
Exercises 1 – 45 EOO

14. Integration as an Accumulation Process
Consider the area under the curve y = sin x
Think of integrating as an accumulation of the areas of the rectangles from 0 to b b

15. 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]

16. Try It Out
Find the accumulation function for
Evaluate
F(0)
F(4)
F(6)

17. Applications
The surface of a machine part is the region between the graphs of y1 = |x| and y2 = 0.08x2 +k
Determine the value for k if the two functions are tangent to one another
Find the area of the surface of the machine part

18. Assignments B
Lesson 7.1B
Page 453
Exercises 57 – 65 odd, 85, 88