Area between two curves
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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


#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)


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