6.1– Areas Between Curves

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In Calculus I we learned how to find the area under a

curve bounded by the x-axis.

In this section we are going to learn how to find areas

between curves.

Example →

3

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

( *) ( *)i if x g x

AREAS BETWEEN CURVES

Consider area bounded by the functions below.

The width of the rectangle will be Dx and

Since that change will be

very small we will call it dx,

and the height will be

5

2( ) 5f x x

( ) 2g x x

1 2

2

2

( ) ( )

5 ( 2)

7

f x g x y y

x x

x x

1y

2y

http://www.slu.edu/classes/maymk/banchoff/AreaBetweenCurves.html

Then the area of the rectangle will be

So the area between

the curves will be the

sum of all rectangles

That is

a, and b are ????

6

2( ) 5f x x

( ) 2g x x

2 7length width x x dx

1y

2y 2 7

b

a

x x dx

Formula for the area between curves:

6.1 Areas Between Curves 7

The area of the region bounded by

the curves y=f(x) and y=g(x) and

the lines x=a and x=b where f and

g are continuous and f(x) g(x)

for all x in [a,b] is given by

Areab

af x g x dx

y = f(x)

y = g(x)

We can find the area

between the two

functions if we divide

it in two pieces

So the total area would

be

Sometimes we will have to find areas that look like the

one below ▼

8

2y x

y x y x

2y x A1

A2

1 2

2 4

0 2 2

A A

x dx x x dx

9

There is however an easier method;

Instead of using Vertical Strips we could use

Horizontal Strips, which would give us

22

02 y y dy

y x

2y x

dy

y x

2y x

2y x

2y x

2

2 3

0

1 12

2 3y y y

82 4

3

10

3

10

1

Decide on vertical or horizontal strips. (Pick whichever is easier to write

formulas for the length of the strip, and/or whichever will let you

integrate fewer times.)

Sketch the curves.

2

3 Write an expression for the area of the strip.

(If the width is dx, the length must be in terms of x.

If the width is dy, the length must be in terms of y.

4 Find the limits of integration. (If using dx, the limits are x values; if

using dy, the limits are y values.)

5 Integrate to find area.

11

Sketch the region enclosed by the given curves. Decide

whether to integrate with respect to x or y. Find the area

of the region.

2

2

1. 1, 9 , 1, 2

2. sin , , 0, / 2

3. tan , 2sin , / 3 / 3

4. 4 12,

5. cos , 1 cos , 0

x

y x y x x x

y x y e x x

y x y x x

x y x y

y x y x x

12

Evaluate the integral and interpret it as the area of

a region. Sketch the region.

Solutions

/2

0

4

0

6. | sin cos 2 |

7. | 2 |

x x dx

x x dx

13

Use a graph to find the x-coordinates of the points of

intersection of the given curves. Then find (possibly

approximately) the area of the region bounded by

the curves.

2 4

2

2 3

8. sin( ),

9. , 2

10. 3 2 , 3 4

x

y x x y x

y e y x

y x x y x x

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