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4.3 Riemann Sums and Definite Integrals

4.3 Riemann Sums and Definite Integrals. Objectives. Understand the definition of a Riemann sum. Evaluate a definite integral using limits. Evaluate a definite integral using properties of definite integrals. Riemann Sums.

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4.3 Riemann Sums and Definite Integrals

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  1. 4.3 Riemann Sums and Definite Integrals

  2. Objectives • Understand the definition of a Riemann sum. • Evaluate a definite integral using limits. • Evaluate a definite integral using properties of definite integrals.

  3. Riemann Sums When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. The width of a rectangle is called a subinterval. The entire interval is called the partition. subinterval partition Subintervals do not all have to be the same size.

  4. Riemann Sums Riemann Sums: • Add areas of rectangles to estimate area. • Rectangle widths don’t have to be the same. • 3 basic types: • Left (use f(left endpoint) as height) • Right (use f(right endpoint) as height) • Midpoint (use f(midpoint) as height)

  5. If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by . As gets smaller, the approximation for the area gets better. subinterval partition Norm of the Partitional

  6. Definition of a Definite Integral If f is defined on [a,b] and the limit exists then f is integrable on [a,b] and the limit is denoted by discrete continuous Longest rectangle width 0 # rectangles  ∞

  7. Definite Integral Notation Leibniz introduced the simpler notation for the definite integral: Note that the very small change in x becomes dx.

  8. Theorem 4.4: Continuity Implies Integrability If a function f is continuous of [a,b], then f is integrable on [a,b].

  9. Example Evaluate the definite integral Remember: Why is it negative?

  10. Theorem If fis continuous and nonnegative on [a,b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is given by

  11. Example Consider the region bounded by the graph of f(x)=4x-x2 and the x-axis.

  12. Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.

  13. Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.

  14. Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.

  15. Properties of Definite Integrals

  16. More Properties

  17. Homework 4.3 (page 278) # 13 – 43 odd 47, 53

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