The Area Between Two Curves

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The Area Between Two Curves - PowerPoint PPT Presentation

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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The Area Between Two Curves

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The Area Between Two Curves

Lesson 6.1

What If … ?
• We want to find the area betweenf(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)

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]

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
• 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 absolutevalue 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
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
• We could make these graphs as functions of y
• Now each slice isy by (k(y) – j(y))
Practice
• Determine the region bounded between the given curves
• Find the area of the region
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
• 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

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