The Area Between Two Curves

Lesson 6.1

What If … ?

• We want to find the area between f(x) and g(x) ?

• Any ideas?

When f(x) < 0

• Consider taking the definite integral for the function shown below.

• The integral gives a negative area (!?) We need to think of this in a different way

a b

f(x)

( )b

a

f x dx

Another Problem

• What about the area between the curve and the x-axis for y = x3

• What do you get forthe integral?

• Since this makes no sense – we need another way to look at it

23

2

x dx

Recall our look at odd functions on the interval [-a, a]

Solution

• We can use one of the properties of integrals

• We will integrate separately for -2 < x < 0 and 0 < x < 2

( ) ( ) ( )b c b

a a c

f x dx f x dx f x dx

2 0 23 3 3

2 2 0

x dx x dx x dx

We take the absolute value for the interval

which would give us a negative area.

We take the absolute value for the interval

which would give us a negative area.

General Solution

• When determining the area between a function and the x-axis Graph the function first Note the zeros of the function Split the function into portions

where f(x) > 0 and f(x) < 0 Where f(x) < 0, take

absolute value of the definite integral

Try This!

• Find the area between the function h(x)=x2 + x – 6 and the x-axis Note that we are not given the limits of

integration We must determine zeros

to find limits Also must take absolute

value of the integral sincespecified interval has f(x) < 0

Area Between Two Curves

• Consider the region betweenf(x) = x2 – 4 and g(x) = 8 – 2x2

• Must graph to determine limits

• Now consider function insideintegral Height of a slice is g(x) – f(x)

So the integral is 2

2

( ) ( )g x f x dx

The Area of a Shark Fin

• Consider the region enclosed by

• Again, we must split the region into two parts 0 < x < 1 and 1 < x < 9

( ) 9 9 ( ) 9f x x g x x x axis

Slicing the Shark the Other Way

• We could make these graphs as functions of y

• Now each slice isy by (k(y) – j(y))

( ) 9 9 ( ) 9f x x g x x x axis

2 21( ) 9 ( ) 9

9j y x y and k y x y

3

0

( ) ( )k y j y dy

Practice

• Determine the region bounded between the given curves

• Find the area of the region

2 6y x y x

Horizontal Slices

• Given these two equations, determine the area of the region bounded by the two curves Note they are x in terms of y

2

2

8x y

x y

Assignments A

• Lesson 7.1A

• Page 452

• Exercises 1 – 45 EOO

Integration as an Accumulation Process

• Consider the area under the curve y = sin x

• Think of integrating as an accumulation of the areas of the rectangles from 0 to b

b

0

sinb

x dx

Integration as an Accumulation Process

• We can think of this as a function of b

• This gives us the accumulated area under the curve on the interval [0, b]

00

( ) sin cos ( ) cos 1b

bA b x dx x b

Try It Out

• Find the accumulation function for

• Evaluate F(0) F(4) F(6)

2

0

1( ) 2

2

x

F x t dt

Applications

• The surface of a machine part is the region between the graphs of y1 = |x| and y2 = 0.08x2 +k

• Determine the value for k if the two functions are tangent to one another

• Find the area of the surface of the machine part

Assignments B

• Lesson 7.1B

• Page 453

• Exercises 57 – 65 odd, 85, 88