The Area Between Two Curves

1
The Area Between Two Curves

• Lesson 6.1

2
What If ?

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

3
When f(x) lt 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)
4
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

Recall our look at odd functions on the interval
-a, a
5
Solution

• We can use one of the properties of integrals
• We will integrate separately for -2 lt x lt 0 and
  0 lt x lt 2

6
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) gt 0
  and f(x) lt 0
• Where f(x) lt 0, take absolute value of the
  definite integral

7
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 absolutevalue of the integral
  sincespecified interval has f(x) lt 0

8
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

9
The Area of a Shark Fin

• Consider the region enclosed by
• Again, we must split the region into two parts
• 0 lt x lt 1 and 1 lt x lt 9

10
Slicing the Shark the Other Way

• We could make these graphs as functions of y
• Now each slice is?y by (k(y) j(y))

11
Practice

• Determine the region bounded between the given
  curves
• Find the area of the region

12
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

13
Assignments A

• Lesson 7.1A
• Page 452
• Exercises 1 45 EOO

14
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
15
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

16
Try It Out

• Find the accumulation function for
• Evaluate
• F(0)
• F(4)
• F(6)

17
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

18
Assignments B

• Lesson 7.1B
• Page 453
• Exercises 57 65 odd, 85, 88