1. 8-2 Areas in the plane

2. Area Between Curves
To find the area between a curve and the x-axis, we can simply integrate.
Now we will learn about area between two curves.
The idea is still the same. – Partition & sum as rectangle widths go to 0.
Def: Area Between Curves
If f and g are continuous with f (x)  g (x) through [a, b], the area between curves f (x) and g (x) from a to b is the integral of [ f – g ] from a to b.
**Surprise! It is very helpful to sketch the graphs first!**

3. Ex 1) Find the area of the region between y = sec2 x and
y = sin x from x = 0 to
y = sec2 x
y = sin x

4. Area enclosed by Intersecting Curves
*We need to find the intersection points.*
Ex 2) Find the area of the region enclosed by the parabola y = 2 – x2 and the line y = – x.
Intersect at: 2 – x2 = –x
0 = x2 – x – 2
0 = (x – 2)(x + 1)
x = 2, –1
y = 2 – x2
y = – x

5. Using a Calculator
Ex 3) Find the area of the region enclosed by the graphs of y = 2cos x and y = x2 – 1
Enter eqtns under Y1 and Y2
2nd CALC  5: intersect
Enter, enter, enter
Quit
2nd ANS  STO under A
Repeat for 2nd intersection point and store under B
y = 2cos x
y = x2 – 1
Math 9: fnInt
 4.995
Y1 – Y2

6. Partitioning a Region
Ex 4) Find the area of the region R in the first quadrant that is bounded above by and below by the x-axis and the line y = x – 2.
I + II II I

7. Higher x-values (on top)
Sometimes it’s easier to integrate with respect to y instead of with respect to x.
Ex 5) Find the area of the region in Ex 4 by integrating with respect to y.
Higher x-values (on top)

8. Using Geometry
Ex 7) Find the area of the region in Ex 4 by subtracting the area of the triangular region from the area under the square root curve.
Whole area under curve – triangle

9. Ex 6) Find the area of the region enclosed by the graphs of
y = x3 and x = y2 – 2.
(Use any strategy you wish – wrt x or wrt y)
If by x, have to partition area
at x = a.
x = a
If by y, don’t have to partition
Find intersection points for upper and lower bounds
 4.21

10. homework
Pg. 390 # 3, 17, 21, 34, 35
Pg. 399 # 3–45 (mult of 3)