0 calc7-1

What’s the definite integral used for?

= -

Area of region between f and g

= Area of regionunder f(x)

- Area of regionunder g(x)

f (x) g(x) a

b

dx b

a

dxxf )( b

a

dxxg )(

f

g

f

g

f

g

7.1 Areas Between CurvesTo find the area:• divide the area into n strips of equal width• approximate the ith strip by a rectangle with base Δx and height f(xi) – g(xi).• the sum of the rectangle areas is a good approximation• the approximation is getting better as n→∞.

y = f(x)

y = g(x)

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

adxxgxfA )]()([

To find the area between 2 curves(along the x-axis)

• Sketch a graph (if you can)• Draw a representative rectangle to determine

the upper and lower curves.• Use the formula:

Area = Top curve – bottom curve

A f (x) g(x) a

b

dx

Ex. Find the area of the region bounded by the graphs of f(x) = x2 + 2, g(x) = -x, x = 0, and x = 1 .

Area = Top curve – bottom curve

A f (x) g(x) a

b

dx

(x 2 2) ( x) 0

1

dx

1

0

23

223

xxx6

17221

31

Find the area of the region bounded by the graphs off(x) = 2 – x2 and g(x) = x

First, set f(x) = g(x) tofind their points of intersection.

2 – x2 = x0 = x2 + x - 20 = (x + 2)(x – 1)

x = -2 and x = 1

2 x 2 x 2

1

dx fnInt(2 – x2 – x, x, -2, 1) 29

Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and g(x) = -x2 + 2x

Again, set f(x) = g(x)to find their points of intersection.

3x3 – x2 – 10x = -x2 + 2x3x3 – 12x = 0

3x(x2 – 4) = 0

x = 0 , -2 , 2Note that the two graphs switch at the origin.

Now, set up the two integrals and solve.

(3x 3 12x)dx ( 30

2

2

0

x 3 12x)dx

24

f (x) g(x) 2

0

dx g(x) f (x) 0

2

dx

dxcurvebottomcurvetopAx

x

])()[(2

1

dyurvectflecurverightAy

y

])()[(2

1

1. Find the area of the region bounded by the graphs of , , x = -2 , and x = 2.

2.Find the area of the region bounded by the graphs of , and .

236 xxy 92 xy

52 xy xy 1

Find the area of the region bounded by the graphs of x = 3 – y2 and y = x - 1

When you integrate with respect to Y:

•Your functions must all be in terms of y•Your variable of integration changes to “dy”•The formula for Area between curves becomes:

dyurvectflecurverightAb

a

])()[(

Find the area of the region bounded by the graphs of x = 3 – y2 and y = x - 1

A (3 y 2) (y 1) 2

1

dy

29

Area = Right - Left

Find the area of the region bounded by the graphs of x = y2 and y = x - 2