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Area Between Two Curves

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 Between Two Curves

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  1. Area Between Two Curves 7.1

  2. 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….

  3. Area between two curves (Graph)

  4. 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.

  5. It’s possible for the upper and lower boundaries to intersect at a point on the left and/or right sides. If this happens we will have to determine the points of intersection to obtain the limits of our integration.

  6. Example 2 • Find the area of the region that is enclosed between the curves y = x² and y = x + 6

  7. It’s possible for the upper and lower boundary to consist of two or more curves, in which case we can subdivide the region into smaller pieces to find the area

  8. Example 3 • Find the area of the region that is enclosed between the curves x = y² and y = x – 2 • What’s different about this question?

  9. 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

  10. #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

  11. Practice • Sketch the region enclosed by and then find the area.

  12. Practice • Pg. 448 (1 – 23 odd)

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