area between two curves objective: to find the area between two curves

Area Between Two Curves

Objective: To find the area between two curves.

Riemann Sums

• Let’s review what a Riemann Sum is:


• First Area Problem: Suppose that f and g are continuous functions on an interval [a, b] and f(x) > g(x) for a < x < b.

• This means that the curve y = f(x) lies above the curve y = g(x) and that the two can touch but never cross.

• Find the area A of the region bounded above by y = f(x), below by y = g(x), and on the sides by the lines x = a and x = b.


• To solve this problem, we divide the interval [a, b] into n subintervals, which has the effect of subdividing the region into n strips. If we assume that the width of the kth strip is , then the area of the strip can be approximated by the area of a rectangle of width and height , where is a point in the kth subinterval.

kx

kx )()( **kk xgxf *

kx


• Adding these approximations yields the following Riemann Sum that approximates the area A:

n

kkkk xxgxfA

1

** )]()([


• Adding these approximations yields the following Riemann Sum that approximates the area A:

• Taking the limit as n increases and the widths of the subintervals approach zero yields the following definite integral for the area A between the curves:

n

kkkk xxgxfA

1

** )]()([

b

a

n

kkkk

xdxxgxfxxgxfA )]()([)]()([lim

1

**

0max


• 7.1.2 Area Formula• If f and g are continuous functions on the interval

[a, b], and if f(x) > g(x) for all x in [a, b], then the area of the region bounded above by y = f(x) and below by y = g(x), on the left by the line x = a, and on the right by the line x = b is

b

a

dxxgxfA )]()([

Example 1

• Find the area of the region bounded above by y = x + 6, bounded below by y = x2, and bounded on the sides by the lines x = 0 and x = 2.

Example 1

• Find the area of the region bounded above by y = x + 6, bounded below by y = x2, and bounded on the sides by the lines x = 0 and x = 2.

3

340

3

34

36

2])6[(

2

0

322

0

2

x

xx

dxxxA

Example 2

• Find the area of the region that is enclosed between the curves y = x2 and y = x + 6.

Example 2


• Looking at the graph, we see that y = x2 is the lower bound and y = x + 6 is the upper bound. We need to find the points of intersection to find a and b. We will do this with our calculator.

Example 2


• Now, we integrate to find the answer.

6

125

3

22

2

27

36

2])6[(

3

2

323

2

2

xx

xdxxxA

Area

• In the case where both f and g are nonnegative on the interval [a, b], the area A between the curves can be obtained by subtracting the area under y = g(x) from the area under y = f(x).

Example 4

• Find the area of the region enclosed by x = y2 and y = x – 2.

Example 4


• The situation that makes this problem different is that the bottom curve is not the same everywhere. We need to look at this as two separate areas and integrate twice.

Example 4


• The top curve is always , but the bottom curve is from 0-1, and it is y = x – 2 from 1-4. The two integrals will be:

xy

xy

1

0

1 )]([ dxxxA 4

1

2 )]2([ dxxxA

Example 4


3

4

3

4)]([

1

0

1

0

2/3 xdxxx

6

192

23

2)]2([

4

1

22/3

4

1

x

xxdxxx

2

9

6

19

3

421 AA

Reversing the Rolls of x and y

• Sometimes it is possible to avoid splitting a region into parts by integrating with respect to y rather than x. We will now look at this situation.

Second Area Problem

• 7.1.3 Suppose that w and v are continuous functions of y on an interval [c, d] and that

for

• This means that lies to the right of and that the two curves can touch but never cross.

dyc )()( yvyw

)(ywx

)(yvx

Area Formula

• 7.1.4 If w and v are continuous functions and if for all y in [c, d], then the area of the

region bounded on the left by , on the right by below by y = c, and above by y = d is )(ywx

)()( yvyw

)(yvx

d

c

dyyvywA )]()([

Example 5

• Find the area of the region enclosed by x = y2 and y = x – 2, integrating with respect to y.

Example 5


• First, we need to solve each equation for x to put it in terms of y. We also need to find the bounds in terms of y.• x = y2

• x = y + 2• c = -1, d = 2

Example 5


• This leads us to the integral:

2

9

32

2

2

1

32

yy

y

2

1

2 ])2[( dyyy

Under vs. Between

• When we found the area under a curve, we were only dealing with one curve and it was possible to have what we called “negative area”. Now, with the area between two curves, we will always have two curves and the area will always be positive.

Under vs. Between

• When we found the area under a curve, we were only dealing with one curve and it was possible to have what we called “negative area”. Now, with the area between two curves, we will always have two curves and the area will always be positive.

• Sometimes, the second curve will be the x or y-axis. It may be the top curve or the bottom curve. For some of these, we will need to use our knowledge of piecewise functions to solve.

Example 6

• Find the area between the curves and y = 0 from . ]2,0[

xy sin

Example 6

• Find the area between the curves and y = 0 from .

• From , the top curve is y = sinx and the bottom curve is y = 0.

• From , the top curve is y = 0 and the bottom curve is y = sinx.

]2,0[ xy sin

],0[

]2,[

Example 6

• Find the area between the curves and y = 0 from .]2,0[

xy sin

2)1(1cos]0[sin00

xdxx

2)1(1cos]sin0[22

xdxx

Homework

• Section 6.1

• 1-19 odd

• 35

area between two curves objective: to find the area between two curves

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