Area of a Region Between 2 Curves

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Area of a Region Between 2 Curves

• Section 6.1

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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

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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

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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

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g
f
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Area Between Curves
Find the area of the shaded region
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Area Between Curves
Find the area of the shaded region
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Area Between Curves
In general
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Area Between Curves
In general
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1. Find the area of the shaded region.
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1. Find the area of the shaded region.
Find intersection points first.
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2. Sketch the region represented by
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2. Sketch the region represented by
top
bottom
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• Representative rectangles are sometimes used.
  A vertical rectangle (of width ) implies
  integration with respect to x, whereas a
  horizontal rectangle (of width ) implies
  integration with respect to y.

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• If 2 curves intersect at more that 2 points,
  then to find the area of the region between the
  curves, you must find all points of intersection
  and check to see which curve is above the other
  in each interval determined by these point.

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In general, to determine the area between 2
curves, you can use
in variable x -vertical rectangles
in variable y - horizontal rectangles
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Area Between Two Curves

• Sketch
• Determine which curve is on top
• Determine a and b
• Integrate!

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Example
Find the area trapped between y  x 2 and y  1.
y  1 y  x 2
Top curve Bottom curve
Solve for a and b
x2  1, so x  1 and x  1.
The correct integral then is
Evaluate
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Example
Set up an integral to find the area trapped
between y  x and in the region for which x
is between 2 and 5.
Can we do this exactly?
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Example 6  Determine the area of the region
enclosed by                             
 and                    .
 
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The area is,                                      
                                                
                                         
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1. Find the area of the region bounded by
No calculator.
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1. Find the area of the region bounded by
No calculator.
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2. Use your calculator to find the area between
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2. Use your calculator to find the area between
A
Intersection points
C
B
Store these values as the respective letter!
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1. Find the area of the region bounded by
No calculator.
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1. Find the area of the region bounded by
No calculator.
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2. Find the area of the triangle with vertices
A(2,-3), B(4,6), and C(6,1).
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2. Find the area of the triangle with vertices
A(2,-3), B(4,6), and C(6,1).
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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

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Try It Out

• Find the accumulation function for
• Evaluate
• F(0) 0
• F(4) 3
• F(6) 16

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AP QUESTIONS
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