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Area Between Two Curves. 7.1. Area Formula. If f and g are continuous functions on the interval [a, b], and if f(x) > g(x) for all x in [a, b], then the area of the region bounded above by y = f(x), below by y = g(x), on the left by x = a and on the right by x = b is…. .

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area formula
Area Formula
  • If f and g are continuous functions on the interval [a, b], and if f(x) > g(x) for all x in [a, b], then the area of the region bounded above by y = f(x), below by y = g(x), on the left by x = a and on the right by x = b is….
example 1
Example 1
  • Find the area of the region bounded above by y = x + 6 and below by y = x², and bounded on the sides by the lines x = 0 and x = 2.
example 11
Example 1
  • Find the area of the region bounded above by y = x + 6 and below by y = x², and bounded on the sides by the lines x = 0 and x = 2.
example 2
Example 2
  • Find the area of the region that is enclosed between the curves y = x² and y = x + 6
example 21
Example 2
  • Find the area of the region that is enclosed between the curves y = x² and y = x + 6
example 3
Example 3
  • Find the area enclosed by the curves

and

example 31
Example 3
  • Find the area enclosed by the curves

and

practice
Practice
  • Sketch the region enclosed by and then find the area.
practice1
Practice
  • Sketch the region enclosed by and then find the area.
example 4
Example 4
  • Find the area of the region that is enclosed between the curves x = y² and y = x – 2
  • What’s different about this question?
example 41
Example 4
  • Find the area of the region that is enclosed between the curves x = y² and y = x – 2
  • What’s different about this question?
two ways to solve
Two ways to solve…..

#1

  • Subdivide the regions
  • Equation of top graph – equation of bottom graph
  • Equations must be solved in terms of y and bound are determined by x values
slide15

#2

  • Reverse the roles of x and y
    • When reversing roles you always subtract the graph on the right – graph on left
    • Must solve equations in terms of x and gets bounds in terms of y
example 5
Example 5
  • Find the area of the region enclosed by the curves and x = 1.
example 6
Example 6
  • Find the area of the region enclosed by the curves and
practice2
Practice
  • Pg. 448 (1 – 23 odd)