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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.

Using definite integrals to find areas of regions that lie between the graphs of two functions.

1.
6

APPLICATIONS OF INTEGRATION

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.

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.

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.

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].

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 *) .

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 .

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 → ∞.

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.

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

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

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

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.

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.

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

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.

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

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

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.

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.

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.

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).

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

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

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.

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.

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.

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.

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

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

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.

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.

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

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

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 .

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

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, . . .

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 +. . .

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.

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, . . . .

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.

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).

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.

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

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

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

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.

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.

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

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.

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

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

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.

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.