1. Area

2. Back to Area:
We can calculate the area between the x-axis and a continuous function f on the interval [a,b] using the definite integral:
Where f(xi*) is the height of a rectangle and ∆x is the width of that rectangle. {(b-a)/n (n is the number of rectangles)}
Remember that the area above the axis is positive and the area below is negative.

3. Set up the integral needed to find the area of the region bounded by: and the x-axis.

4. Set up the integral needed to find the area of the region bounded by: , the x-axis on [0,2].

5. Area bounded by two curves
Suppose you have 2 curves, y = f(x) and y = g(x)
Area under f is:
Area under g is:

6. Superimposing the graphs, we look at the area bounded by the two functions:
(top - bottom)*∆x

7. The area bounded by two functions can be found:

8. Find the area of the region between the two functions: and
Bounds?
[-1,2]
Top Function?
Bottom Function?
Area?
= 9

9. Find the area bounded by the curves: and
Solve for bounds:

10. Find the area bounded by the curves: and
Sketch the graph:
(top - bottom)*∆x

11. Find the area of the region determined by the curves: and
Bounds?
In terms of y: [-2,4]
Points (-1,-2) & (5,4)
Graph?
Solve for y:

12. Find the area of the region determined by the curves: and
Need 2 Integrals!
One from -3 to -1 and the other from -1 to 5.
Area?

13. Horizontal Cut instead:
Bounds?
In terms of y: [-2,4]
Right Function?
Left Function?
Area?
= 18

14. In General:
Vertical Cut:
Horizontal Cut:

15. Find the Area of the Region bounded by and
Bounds?
[0,1]
Top Function?
Bottom Function?
Area?

16. Find the Area of the Region bounded by , , and the y-axis
Bounds?
[0,π/4] and [π/4, π/2]
[0,π/4]
Top Function?
Bottom Function?
Area?

17. Find the Area of the Region bounded by , , and the y-axis
Bounds?
[0,π/4] and [π/4, π/2]
Top Function?
Bottom Function?
Area?

18. Find the area of the Region bounded by
Bounds?
Interval is from -2, 5
Functions intersect at x = -1 and x = 3
Graph?
Top function switches 3 times!
This calculation requires 3 integrals!

19. Find the area of the Region bounded by

20. Find T so the area between y = x2 and y = T is 1/2.
Bounds?
Top Function?
Bottom Function?
Area?
Taking advantage of Symmetry
Area must equal 1/2:
Ans: