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Definite Integral

Definite Integral. Two main points of view: Integral as (signed) area under curve Integral as antiderivative Riemann integral. Area Under Curves. Find area under curve between x=a and x=b by taking limits: First subdivide [a,b] into n equal subintervals

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Definite Integral

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  1. Definite Integral • Two main points of view: • Integral as (signed) area under curve • Integral as antiderivative • Riemann integral

  2. Area Under Curves • Find area under curve between x=a and x=b by taking limits: • First subdivide [a,b] into n equal subintervals • Choose point xk in each subinterval and compute area of rectangle with height f(xk) and base the k-th subinterval • Let An be sum of areas of rectangle • Area under curve is limit of An

  3. Picture

  4. Explicit Example • Area under curve y = x2 over the interval[0,1]

  5. n subintervals of [0,1] have endpoints 0, 1/n, 2/n, … , (n - 1)/n, 1 • Value of f(x) = x2 at right endpoint is (1/n)2, (2/n)2, … , ( (n-1)/n)2, 1 • An = [ (1/n)2 + (2/n)2 + … + 12] / n • A1 = 1, A4= 0.46875, A10=0.385, A100 = 0.33835, A100,000 = 0.333338

  6. Antiderivatives • Observation of Newton and Leibniz: • Let A(x) be area under curve y = f(x) between a and x, then A(x) = f(x)

  7. Why?

  8. Moral • A(x) is an antiderivative of f: A(x) = f(x) • Antiderivatives differ by a constant; need to choose right one to get area • Example: x3/3 is an antiderivative of x2 . But so is x3/3 + C. • To get area from x=0 to x=1 notice that A(0)=0 so want to choose C=0 • Conclude that A(x) = x3/3 for this example

  9. Examples • Use the antiderivative method to compute: • Area under curve y = 4x + 2 between x=0 and x=2 • Area under curve y=ex between x = 1 and x = 3 • Area under curve y=1/x between x=2 and x=100

  10. Definite Integral notation • Assuming that f is continuous and f(x) 0 on [a,b], we denote the area under the curve y=f(x) between x=a and x=b by

  11. Signed area under curve • Consider the graph of f(x) for x in [a,b]. Suppose that A+is the area of the region above the x-axis, and A- is the area under the x-axis. • Then

  12. Examples • Find the values of the following integrals by interpreting them as the signed area of an appropriately chosen region

  13. Sigma notation

  14. Definition of Riemann integral

  15. Properties of the Riemann integral

  16. Examples

  17. Comparison Properties

  18. Examples • Verify each inequality without evaluating the integrals

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