Section 7.1 – Area of a Region
Between Two Curves
The circle below is inscribed into a square:
Calculator
White Board Challenge
20 cm
What is the shaded area?
400  100  85.841 cm
2
Calculator
White Board Challenge
Find the area of the region bounded by the function
below and the x-axis between x = 1 to x = 6:
f  x   0.1 x  5   2
2
 f  x  dx
   0.1 x  5 
6
1
6
1
 12.16
2

 2 dx
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.
Find the area of the region between y = sec2x
and y = sin x from x = 0 to x = π/4:
y  sec 2 x
Subtracting the bottom
area from the top, leaves
only the area
in-between.
Area between the curves 

 4
0

y  sin x
4

TOP
sec x dx  
 4
0

Calculator
White Board Challenge
2
2
2
 sec
TOP
2
 4
0
BOTTOM
sin x dx
x  sin x  dx
BOTTOM
In this example, all of the
area was above the x-axis.
Does the same process work
for “negative” area?
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.
f  x
Area between the curves

Between
(Positive)
a
Between
(Negative)
b
a
BOTTOM
f  x  dx   g  x  dx
b
a
TOP
BOTTOM
f
x

g
x
dx







a 
b
b
g  x
Must be positive!
TOP
THE SAME!
In this example, one area was
positive and one was negative. Does
the same process work if both areas
are negative?
Area Between Two Curves:
Negative Area Only
Find the area of the region between the two
curves from x = a to x = b:
a
f  x
Outside
b
Area between the curves
TOP
BOTTOM
 f  x  dx   g  x  dx
b
(Counted Twice)
a
Between
(Negative)
b
a
TOP
BOTTOM
 f  x   g  x   dx
a

b
THE SAME!
Subtracting the negative
area switches it to adding
a positive version AND
cancels the outside area.
g  x
In this example, both areas were
negative. Now we can apply the
three scenarios to any two curves.
Area Between Two Curves:
A Mix
Find the area of the region between the two
curves from x = a to x = b:
f  x
Area between the curves

NEG-NEG
b
a
TOP
f  x  dx   g  x  dx
POS-POS
POS-NEG
TOP
BOTTOM
f
x

g
x
dx







a 
b
g  x
b
a
b
a
BOTTOM
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:
A    f  x   g  x   dx
a
b
TOP
BOTTOM
Reminder: Riemann Sums
Recall that the integral is a limit of Riemann Sums:
Area 
xk
lim
n

max xk 0
f  x
k 1
 f  xk*   g  xk*   xk


f  xk*   g  xk* 
a
g  x
b
   f  x   g  x   dx
a
b
Example 1
Find the area of the region between the graphs of
the functions
f  x   x  4 x  10, g  x   4 x  x , 1  x  3
2
2
Sketch a Graph
Find the Boundaries/Intersections
x  1,3
Base = dx
Height
=f–g
Make Generic
“Riemann”
Rectangle(s)
Integrate the Area of Each
Generic Rectangle
2
2


x

4
x

10

4
x

x




1 
 dx
16

3
3
Example 2
Find the area of the region enclosed by the
parabolas y = x2 and y = 2x – x2.
Sketch a Graph
Base = dx
Height =
(2x–x2)–(x2)
Make Generic
“Riemann”
Rectangle(s)
Find the Boundaries/Intersections
x2  2x  x2
2 x2  2 x  0
2 x  x  1  0
x  0,1
Integrate the Area of Each
Generic Rectangle
2
2


2x

x

x




0 
 dx
1

3
1
Example 3
Find the area of the region bounded by the graphs
y = 8/x2, y = 8x, and y = x.
Sketch a Graph
Height
Base
= 8/x2-x
= dx Base
Height
= dx
= 8x-x
Find the Boundaries/Intersections
8x  x
x0
8 x  x82
x 1
Integrate the Area of Each
Generic Rectangle
2
8
2
x
1
 8 x  x  dx   
1
0
Make Generic “Riemann”
Rectangle(s)
x  x82
x2
6

 x dx
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
Base
Base
= dx
= dx
Find the Boundaries/Intersections
x  0, 2



0
Height = Height =
cos-sin sin-cos
Make Generic 
“Riemann” 2
Rectangle(s)
sin x  cos x
x  4
Integrate the Area of Each
Generic Rectangle
4
 2
 cos x  sin x  dx    sin x  cos x  dx
 4
 2 2 2
What other Integrals could be used?
 4
2   cos x  sin x  dx (Symmetrical)
0

 2
0
cos x  sin x dx
(Keeps it Positive)
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:
A   f  x   g  x  dx
b
a
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.
“Warm-up”: 1985 Section I
No Calculator
NOW WE CAN DO!
3


x

8

x

8
dx





0 

1
1
4
Calculator
White Board Challenge
Find the area enclosed by the line y = x – 1
2
and the parabola y = 2x + 6.
y  2x  6
 
1
3

2 x  6   2 x  6 dx
1
y  x 1
y   2x  6
   2 x  6   x  1  dx
3
 18
Example 5
Find the area enclosed by the line y = x – 1 and the
parabola y2 = 2x + 6.
Sometimes Solve for x
Sketch a Graph
y  x 1
x  y 1
y  2x  6
x  12 y 2  3
2
Find the Boundaries/Intersections
1 2
y

1

Base = dy
2 y 3
Height=(y+1)–(1/2y2–3)
0  y2  2 y  8
0   y  4  y  2 
y  2, 4
Integrate the Area of Each
Generic Rectangle
Make Generic “Riemann”
Rectangle(s)
1 2

y

1

2    2 y  3 dy  18
4
White Board Challenge
Using two methods (one with dx and one with dy), find
the area between the x-axis and the two curves:
y x
y  x or x  y 2
&

2
y  x2
x dx  
0
4
2


x   x  2  dx
 103
OR
y  x  2 or x  y  2

2
0
10
y

2

y
d
y

   3
2