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In this section we learn about:
Using definite integrals to find areas of regions that lie between the graphs of two functions.
1.
6
APPLICATIONS OF INTEGRATION
2.
APPLICATIONS OF INTEGRATION
In this chapter, we explore some of the
applications of the definite integral by using it
to compute areas between curves, volumes of
solids, and the work done by a varying force.
The common theme is the following general method—
which is similar to the one used to find areas under
curves.
3.
APPLICATIONS OF INTEGRATION
We break up a quantity Q into a large
number of small parts.
Next, we approximate each small part by a quantity
of the form f ( xi *) x and thus approximate Q by
a Riemann sum.
Then, we take the limit and express Q as an integral.
Finally, we evaluate the integral using the Fundamental
Theorem of Calculus or the Midpoint Rule.
4.
APPLICATIONS OF INTEGRATION
6.1
Areas Between Curves
In this section we learn about:
Using integrals to find areas of regions that lie
between the graphs of two functions.
5.
AREAS BETWEEN CURVES
Consider the region S that lies between two
curves y = f(x) and y = g(x) and between
the vertical lines x = a and x = b.
Here, f and g are
continuous functions
and f(x) ≥ g(x) for all
x in [a, b].
6.
AREAS BETWEEN CURVES
As we did for areas under curves in Section
5.1, we divide S into n strips of equal width
and approximate the i th strip by a rectangle
with base ∆x and height f ( xi *) g ( xi *) .
7.
AREAS BETWEEN CURVES
We could also take all the sample points
to be right endpoints—in which case
xi * xi .
8.
AREAS BETWEEN CURVES
n
The Riemann sum f ( x *) g ( x *) x
i 1
i i
is therefore an approximation to what we
intuitively think of as the area of S.
This approximation appears to become
better and better as n → ∞.
9.
AREAS BETWEEN CURVES Definition 1
Thus, we define the area A of the region S
as the limiting value of the sum of the areas
of these approximating rectangles.
n
A lim f ( xi *) g ( xi *) x
n
i 1
The limit here is the definite integral of f - g.
10.
AREAS BETWEEN CURVES Definition 2
Thus, we have the following formula for area:
The area A of the region bounded by
the curves y = f(x), y = g(x), and the lines
x = a, x = b, where f and g are continuous
and f ( x) g ( x) for all x in [a, b], is:
b
A f x g x dx
a
11.
AREAS BETWEEN CURVES
Notice that, in the special case where
g(x) = 0, S is the region under the graph of f
and our general definition of area reduces to
Definition 2 in Section 5.1
12.
AREAS BETWEEN CURVES
Where both f and g are positive, you can see
from the figure why Definition 2 is true:
A area under y f ( x) area under y g ( x)
b b
f ( x)dx g ( x) dx
a a
b
f ( x) g ( x) dx
a
13.
AREAS BETWEEN CURVES Example 1
Find the area of the region bounded
above by y = ex, bounded below by
y = x, and bounded on the sides by
x = 0 and x = 1.
14.
AREAS BETWEEN CURVES Example 1
As shown here, the upper boundary
curve is y = ex and the lower boundary
curve is y = x.
15.
AREAS BETWEEN CURVES Example 1
So, we use the area formula with y = ex,
g(x) = x, a = 0, and b = 1:
1
2 1
A e x dx e
x x 1
2 x
0 0
1
e 1 e 1.5
2
16.
AREAS BETWEEN CURVES
Here, we drew a typical approximating
rectangle with width ∆x as a reminder of
the procedure by which the area is defined
in Definition 1.
17.
AREAS BETWEEN CURVES
In general, when we set up an integral for
an area, it’s helpful to sketch the region to
identify the top curve yT , the bottom curve yB,
and a typical
18.
AREAS BETWEEN CURVES
Then, the area of a typical rectangle is
(yT - yB) ∆x and the equation
n b
A lim ( yT yB ) x yT yB dx
n a
i 1
summarizes the procedure of adding (in a
limiting sense) the areas of all the typical
19.
AREAS BETWEEN CURVES
Notice that, in the first figure, the left-hand
boundary reduces to a point whereas, in
the other figure, the right-hand boundary
reduces to a point.
20.
AREAS BETWEEN CURVES
In the next example, both the side
boundaries reduce to a point.
So, the first step is to find a and b.
21.
AREAS BETWEEN CURVES Example 2
Find the area of the region
enclosed by the parabolas y = x2 and
y = 2x - x2.
22.
AREAS BETWEEN CURVES Example 2
First, we find the points of intersection of
the parabolas by solving their equations
This gives x2 = 2x - x2, or 2x2 - 2x = 0.
Thus, 2x(x - 1) = 0, so x = 0 or 1.
The points of intersection are (0, 0) and (1, 1).
23.
AREAS BETWEEN CURVES Example 2
From the figure, we see that the top and
bottom boundaries are:
yT = 2x – x2 and yB = x2
24.
AREAS BETWEEN CURVES Example 2
The area of a typical rectangle is
(yT – yB) ∆x = (2x – x2 – x2) ∆x
and the region lies between x = 0 and x = 1.
So, the total area is:
1 1
A 2 x 2 x 2
dx 2 x x dx
2
0 0
2 3 1
x x 1 1 1
2 2
2 3 0 2 3 3
25.
AREAS BETWEEN CURVES
Sometimes, it is difficult—or even impossible
—to find the points of intersection of two
curves exactly.
As shown in the following example, we can
use a graphing calculator or computer to find
approximate values for the intersection points
and then proceed as before.
26.
AREAS BETWEEN CURVES Example 3
Find the approximate area of the region
2
bounded by the curves y x x 1
4
and y x x.
27.
AREAS BETWEEN CURVES Example 3
If we were to try to find the exact intersection
points, we would have to solve the equation
x 4
x x
2
x 1
It looks like a very difficult equation to solve exactly.
In fact, it’s impossible.
28.
AREAS BETWEEN CURVES Example 3
Instead, we use a graphing device to
draw the graphs of the two curves.
One intersection point is the origin. The other is x ≈ 1.18
If greater accuracy
is required,
we could use
Newton’s method
or a rootfinder—if
available on our
graphing device.
29.
AREAS BETWEEN CURVES Example 3
Thus, an approximation to the area
between the curves is:
1.18 x
A 2 x x dx
4
0
x 1
To integrate the first term, we use
the substitution u = x2 + 1.
Then, du = 2x dx, and when x = 1.18,
we have u ≈ 2.39
30.
AREAS BETWEEN CURVES Example 3
2.39 du 1.18
x x dx
4
A 1
2 1
u 0
5 2 1.18
2.39 x x
u
1
5 2 0
5 2
(1.18) (1.18)
2.39 1
5 2
0.785
31.
AREAS BETWEEN CURVES Example 4
The figure shows velocity curves for two cars,
A and B, that start side by side and move
along the same road.
What does the area
between the curves
Use the Midpoint Rule
to estimate it.
32.
AREAS BETWEEN CURVES Example 4
The area under the velocity curve A
represents the distance traveled by car A
during the first 16 seconds.
Similarly, the area
under curve B is
the distance traveled
by car B during that
time period.
33.
AREAS BETWEEN CURVES Example 4
So, the area between these curves—which is
the difference of the areas under the curves—
is the distance between the cars after 16
34.
AREAS BETWEEN CURVES Example 4
We read the velocities
from the graph and
convert them to feet per
5280
1mi /h ft/s
3600
35.
AREAS BETWEEN CURVES Example 4
We use the Midpoint Rule with n = 4
intervals, so that ∆t = 4.
The midpoints of the intervals are t1 2, t2 6,
t3 10, t4and
14 .
36.
AREAS BETWEEN CURVES Example 4
We estimate the distance between the
cars after 16 seconds as follows:
16
(v
0 A vB ) dt t 13 23 28 29
4(93)
372 ft
37.
AREAS BETWEEN CURVES
To find the area between the curves y = f(x)
and y = g(x), where f(x) ≥ g(x) for some values
of x but g(x) ≥ f(x) for other values of x, split
the given region S into several regions S1,
S2, . . . with areas
A1, A2, . . .
38.
AREAS BETWEEN CURVES
Then, we define the area of the region S
to be the sum of the areas of the smaller
regions S1, S2, . . . , that is, A = A1 + A2 +. . .
39.
AREAS BETWEEN CURVES
f ( x) g ( x) when f ( x) g ( x)
f ( x) g ( x)
g ( x) f ( x) when g ( x) f ( x)
we have the following expression for A.
40.
AREAS BETWEEN CURVES Definition 3
The area between the curves y = f(x) and
y = g(x) and between x = a and x = b is:
b
A f ( x) g ( x) dx
a
However, when evaluating the integral, we must still
split it into integrals corresponding to A1, A2, . . . .
41.
AREAS BETWEEN CURVES Example 5
Find the area of the region bounded
by the curves y = sin x, y = cos x,
x = 0, and x = π/2.
42.
AREAS BETWEEN CURVES Example 5
The points of intersection occur when
sin x = cos x, that is, when x = π / 4
(since 0 ≤ x ≤ π / 2).
43.
AREAS BETWEEN CURVES Example 5
Observe that cos x ≥ sin x when
0 ≤ x ≤ π / 4 but sin x ≥ cos x when
π / 4 ≤ x ≤ π / 2.
44.
AREAS BETWEEN CURVES Example 5
So, the required area is:
2
A cos x sin x dx A1 A2
0
4 2
0
cos x sin x dx
4
sin x cos x dx
4 2
sin x cos x 0 cos x sin x 4
1 1 1 1
0 1 0 1
2 2 2 2
2 2 2
45.
AREAS BETWEEN CURVES Example 5
We could have saved some work by noticing
that the region is symmetric about x = π / 4.
4
So, A 2 A1 2 cos x sin x dx
0
46.
AREAS BETWEEN CURVES
Some regions are best treated by
regarding x as a function of y.
If a region is bounded by curves with equations x = f(y),
x = g(y), y = c, and
y = d, where f and g
are continuous and
f(y) ≥ g(y) for c ≤ y ≤ d,
then its area is:
d
A f ( y ) g ( y ) dy
c
47.
AREAS BETWEEN CURVES
If we write xR for the right boundary and xL
for the left boundary, we have:
d
A xR xL dy
c
Here, a typical
approximating rectangle
has dimensions xR - xL
and ∆y.
48.
AREAS BETWEEN CURVES Example 6
Find the area enclosed by
the line y = x - 1 and the parabola
y2 = 2x + 6.
49.
AREAS BETWEEN CURVES Example 6
By solving the two equations, we find that the
points of intersection are (-1, -2) and (5, 4).
We solve the equation of the parabola for x.
From the figure, we notice
that the left and right
boundary curves are:
2
xL y 3
1
2
xR y 1
50.
AREAS BETWEEN CURVES Example 6
We must integrate between
the appropriate y-values, y = -2
and y = 4.
51.
AREAS BETWEEN CURVES Example 6
4
Thus, A 2 xR xL dy
4
y 1
2
1
2 y 3 dy
2
4
1
2 y y 4 dy
2
2
4
3 2
1 y
y
4 y
2 3 2 2
16 (64) 8 16 4
3 2 8 18
52.
AREAS BETWEEN CURVES
In the example, we could have found
the area by integrating with respect to x
instead of y.
However, the calculation is much more
53.
AREAS BETWEEN CURVES
It would have meant splitting the region
in two and computing the areas labeled
A1 and A2.
The method used in
the Example is much
easier.
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