
Integral
calculus 

Applications
of the definite integral

The
area between two curves

The
area between two curves examples







The
area between two curves

If
given are two continuous functions, f
and g
defined over an interval [a,
b],
with g
(x)
<
f (x)
for all x
in
[a,
b],
then the area A
of the region bounded (or enclosed) by these two curves and the
lines x
= a
and
x
= b
is
given by



The
area between two curves examples

Example:
Find the area of the
region bounded by the curve f
(x) = 
x^{2}  2x
and the line g(x)
=
x.

Solution: To
find points of intersections (limits of integration) of the given
functions we solve their equations,


x^{2}  2x =
x
or
x^{2} + 3x = 0, 
x(x
+ 3) = 0,
x_{1} = 0 and
x_{2}
= 3. 
Thus,
the area 





Example:
Find the area of the
region lying to the left of the yaxis
and enclosed by the curves,

f
(x) = (1/3)x^{3} 
3x and g(x) =
x^{2} + 3x.

Solution: To
find points of intersections (limits of integration) of the given curves
we solve their equations,


Thus,
the area 





Example:
Find the area of the
region bounded by the curve y^{2}
=
x and the line y
= 
x
+ 2.

Solution: To
find points of intersections (limits of integration) of the given curve
and the line we solve their

equations,
(
x
+ 2)^{2} =
x
or x^{2}

5x + 4 = 0, 
(x

1)(x

4) = 0,
x_{1} = 1 and
x_{2}
= 4. 
Thus,
the area 




The same area can be calculated by
changing the role of the variables (or the coordinate axes) to get simpler integral expression, therefore












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