1. Section 5.3: Finding the Total Areab
y = f(x) x y
y = g(x)
Shaded Area =
Shaded Area =

2. A) Area Between Two Curves in [a , b]
y = g(x)
y= f(x) a b
y= f(x) a b
y = g(x)
y= f(x) a b
y = g(x)
Area under f(x) =
Area under g(x) =
Area between f(x) and g(x)

3. B) Area Between a Curve and the x-Axis in [a , b]
x-Axis is same as y = 0
y = g(x) a b y x
y = f(x)
y = 0
y = 0
Top Function is y = f(x)
Top Function is y = 0
Bottom Function is y = 0
Bottom Function is y = g(x)
Area under f(x) in [a , b] :
Area under g(x) in [a , b] :

4. y x C) Area Between Intersecting Curves
Example: Find the area between the graph y= x and y = 2x - 1
1) Graph both functions y x
2) Find the points of intersection by equating
both functions:
y = 2x - 1
y = y
y = x2 - 4
x2 - 4 = 2x - 1
x2 - 2x - 3 = 0 a b
(x + 1)(x - 3) = 0
x = -1 , x = +3
3) Area
Top Function is 2x - 1, Bottom Function is x2 - 4

5. Example: Find the area between the graph y = x2 - 4x + 3 and the x-axis
1) Graph the function with the x-axis (or y = 0) y x
2) Find the points of intersection by equating
both functions:
y = x2 - 4x +3
y = y
x2 - 4x + 3 = 0
(x - 1)(x - 3) = 0 1 3
x = 1 , x = 3
y = 0
3) Area
Top Function is y = 0, Bottom Function is y = x2 - 4x + 3

6. y = x2 - 4 and the x-axis cross each others x = -2, x = 2
D) Area Between Curves With Multiple Points of Intersections (Crossing Curves)
Example: Find the area enclosed by the x-axis, y = x2 - 4, x = -1 and x = 3
1) Graph the function y = x2 - 4 with the x-axis (or y = 0),
Shade in the region between x = -1 and x = 3 y x
2) Find the points of intersection by equating
both functions:
y= x2 – 4
y = y 3 -1
y = 0
x = 0 or
(x - 2)(x + 2) = 0
x = 2 , x = -2 -2 2
y = x2 - 4 and the x-axis cross each others x = -2, x = 2
A2 = 2.33
3) Area
A1 = 9
= 9 =

7. The graphs cross at x = -1, x = 0 and x = 1
Example: Find the area enclosed by y = x3 and y = x
1) Graph the function y = x3 , and y = x
2) Find the points of intersection by equating
both functions:
y = y y x
y = x3
x3 = x or
x3 - x = 0
y = x
x(x2 - 1) = 0 1 -1
x(x - 1)(x + 1) = 0
x = 0 , x = -1 , x = 1
The graphs cross at x = -1, x = 0 and x = 1
A2 = 0.25
3) Area
A1 = 0.25 =
= 0.5
Note: if you write Area =
, the answer will be zero.