Using Integration to
Determine bounded area
Write the Definite Integral to
determine the shaded area below.
4
ANSWER :
2
x
∫ dx
1
x 3 4 4 3 13
[ ]1 =
−
3
3 3
64 1
=
−
3 3
63
=
= 21
3
Evaluate the Integral without your calculator.
Try another one…..
Write the integral that would
Describe the shaded area.
ANSWER :
2
3
(x
∫ + 3) dx =12.75
−1
Evaluate the Integral on your calculator.
How could you find the area
bounded between f(x), g(x), x=1
and x=3?
How about finding the area under
f(x) between x=1 and x=3……..
……and subtracting the area under
g(x) from x=1 to x=3
Write the Definite Integral to
determine the area bounded
between f(x), g(x), x=1 and x=3.
How about………
3
∫
1
3
3
f (x) dx − ∫ g(x) dx
1
or
∫ [ f (x) − g(x)] dx
1
3
3
∫ f (x) dx − ∫ g(x) dx
1
3
or
1
∫ [ f (x) − g(x)] dx
1
Notice the pattern………….
right boun dary
∫
left boun dary
(top curve − bottom curve) dx
Would the area bounded by f(x) and g(x) be
zero?
b
∫ f (x) dx = +
a
b
∫ g(x) dx = −
a
b
Therefore : ∫ [ f (x) − g(x)] dx =
a
b
b
a
a
∫ f (x) dx − ∫ g(x) dx
=pos – neg = pos
18
When we determine the area
bounded by two functions, the
answer will always be positive!!
Write the Definite Integral to
Determine the area bounded by
h(x) and r(x)
c
Answer:
∫ [h(x) − r(x)] dx
e
Write the Definite Integral to
determine all area bounded by h(x)
and r(x)
Answer:
n
c
e
n
∫ [r(x) − h(x)] dx + ∫ [h(x) − r(x)] dx
Write a Definite Integral to determine the first
quadrant area bound by f(x), the x and y axes.
Evaluate the integral on your calculator.
1.347
ANSWER:
∫
0
f (x)dx = 2.42
Do you understand?? Determine
the area bounded by
y=x2 and y=2x+3.
Do you understand?? Determine
the area bounded by
y=x2 and y=2x+3.
Step 1: Determine the points of intersection:
x2=2x+3
x2-2x-3=0
(x+1)(x-3)=0
x+1=0, x-3=0
x=-1, x=3
Do you understand?? Determine
the area bounded by
y=x2 and y=2x+3.
Now write the integral and solve:
3
2
(2x
+
3
−
x
) dx = 10.667
∫
−1
One more: Determine the area
bounded by –x2+2 and x+1.
.618
ANSWER:
∫
−1.618
[−x 2 + 2 − (x + 1)] dx = 1.863