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Chapter 6 – Applications of Integration

6.1 Areas Between Curves

6.1 Areas Between Curves Dr. Erickson

6.1 Areas Between Curves2

Example

21 2y x

2y x

How can we find the area between these two curves?

We could split the area into several sections, use subtraction and figure it out, but there is an easier way.

Dr. Erickson

6.1 Areas Between Curves3

Example Continued

21 2y x

2y x

Consider a very thin vertical strip.

The length of the strip is:

1 2y y or 22 x x

Since the width of the strip is a very small

change in x, we could call it dx.

Dr. Erickson

6.1 Areas Between Curves4

Example Continued

21 2y x

2y x

1y

2y

1 2y ydx

Since the strip is a long thin rectangle, the area of the strip is:

2length width 2 x x dx

If we add all the strips, we get: 2 2

12 x x dx

Dr. Erickson

6.1 Areas Between Curves5

Example Continued

21 2y x

2y x

2 2

12 x x dx

23 2

1

1 12

3 2x x x

8 1 14 2 2

3 3 2

8 1 16 2

3 3 2

36 16 12 2 3

6

27

6

9

2

2 2

1

92

2x x dx

Dr. Erickson

6.1 Areas Between Curves6

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

Formula for the area between curves:

Areab

af x g x dx

y = f(x)

y = g(x)

Dr. Erickson

6.1 Areas Between Curves7

y x

2y x

y x

2y x

If we try vertical strips, we have to integrate in two parts:

dx

dx

2 4

0 2 2x dx x x dx

We can find the same area using a horizontal strip.

dy Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.y x

2y x

2y x

2y x

Example

Dr. Erickson

6.1 Areas Between Curves8

Example Continued

y x

2y x

We can find the same area using a horizontal strip.

dy

Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.

y x2y x

2y x

2y x

2 2

02 y y dy

length of strip

width of strip

22 3

0

1 12

2 3y y y

82 4

3 10

3

2 2

0

102

3y y dy

Dr. Erickson

6.1 Areas Between Curves9

General Strategy for Area Between Curves: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.

Dr. Erickson

6.1 Areas Between Curves10

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Find the area of the region.

Try These…

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

Dr. Erickson

6.1 Areas Between Curves11

Evaluate the integral and interpret it as the area of a region. Sketch the region.

More Examples:

/2

0

4

0

6. | sin cos 2 |

7. | 2 |

x x dx

x x dx

Dr. Erickson

6.1 Areas Between Curves12

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.

Examples:

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

Dr. Erickson