Area Bounded by Curves
1.5
1.0
sin x
y
0.5
0.0
0
1
2
x
-0.5
cos x
-1.0
-1.5
3
4
5
Area Bounded by Curves
(Integrating w.r.t. x)
Let f and g be continuous functions. Let R be the
region bounded above by y = g(x), below by y = f (x),
on the left by x = a and on the right by x = b. Then R
has area
b
 g x   f x  dx
a
y 8
y  6  x2
6
yx
4
2
6  x  x
2
0
-4
-2
0
2
4
x

x
-2
-4
Area Bounded by Curves
(Integrating w.r.t. y)
Let f and g be continuous functions. Let R be the
region bounded on the right by x = g(y), on the left by
x = f (y), below by y = c and above by y = d. Then R
has area
d
 g  y   f  y  dy
c
y 8
y  6  x2
6
yx



y
4

6 y   6 y
2
 y   
6 y

0
y

-4
-2
0
2
4
x
-2
-4
y 2
x  1 y2
x  2  y2
1
0
0
1
2
3
x
-1
-2
y
4
yx
2
0
-4
-2
0
2
4
6
8
x
-2
x  6  y2
-4
y
8
y  x2
6
4
2
yx
0
0
1
2
3
x
y
4
x  4  y2
3
x  y2
2
1
0
-1
0
-1
-2
-3
-4
1
2
3
4
5
x
y
2
y  1 2 x
y x
1
0
-0.5
0.0
-1
-2
0.5
y  x
1.0
1.5
x
Problems Where The Graph Isn’t Given
Find the area of the region enclosed by the curves.
1. x = 2y2
x = 2 – y2
2. y = x2 – 2x + 1
y = 4 – x2