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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.

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Presentation Transcript
what if
What If … ?
  • We want to find the area betweenf(x) and g(x) ?
  • Any ideas?
when f x 0
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)

another problem
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]

solution

We take the absolute value for the interval which would give us a negative area.

Solution
  • We can use one of the properties of integrals
  • We will integrate separately for -2 < x < 0 and 0 < x < 2
general solution
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
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) < 0
area between two curves
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
the area of a shark fin
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
slicing the shark the other way
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))
practice
Practice
  • Determine the region bounded between the given curves
  • Find the area of the region
horizontal slices
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
assignments a
Assignments A
  • Lesson 7.1A
  • Page 452
  • Exercises 1 – 45 EOO
integration as an accumulation process
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

integration as an accumulation process1
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]
try it out
Try It Out
  • Find the accumulation function for
  • Evaluate
    • F(0)
    • F(4)
    • F(6)
applications
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
Assignments B
  • Lesson 7.1B
  • Page 453
  • Exercises 57 – 65 odd, 85, 88