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## Integrals on the Calculator, discontinuous integrable functions

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**Section 5.2b**Integrals on the Calculator, discontinuous integrable functions**Do Now: Exploration 1 on page 264**It is a fact that With this information, determine the values of the following integrals. Explain your answers (use a graph, when necessary). 1. 4. 2. 5. 3. 6.**Do Now: Exploration 1 on page 264**It is a fact that With this information, determine the values of the following integrals. Explain your answers (use a graph, when necessary). 7. 9. 8. 10. Suppose k is any positive number. Make a conjecture about**A Similar Challenge: #29-38 on p.267-268**Use graphs, your knowledge of area, and the fact that to evaluate the given integrals. 29. 32. 30. 33. 31. 34.**A Similar Challenge: #29-38 on p.267-268**Use graphs, your knowledge of area, and the fact that to evaluate the given integrals. 35. 37. 36. 38.**Integrals on the Calculator**Our modern calculators are good at calculating Riemann sums…our text denotes this function as NINT: We write this statement with an understanding that the right- hand side of the equation is an approximation of the left-hand side…**Integrals on the Calculator**Examples: Evaluate the following integrals numerically.**Discontinuous Integrable Functions**As we already know, a function is not differentiable where it is discontinuous. However, we can integrate functions that have points of discontinuity. Examples… Let’s look at the graph… Find 1 Discontinuity at x = 0!!! –1 1 2 –1 Areas of rectangles: What does our calculator give us on this one???**Discontinuous Integrable Functions**As we already know, a function is not differentiable where it is discontinuous. However, we can integrate functions that have points of discontinuity. Examples… Explain why the given function is not continuous on [0, 3]. What kind of discontinuity occurs? Removable discontinuity at x = 2**Discontinuous Integrable Functions**As we already know, a function is not differentiable where it is discontinuous. However, we can integrate functions that have points of discontinuity. Examples… Use areas to show that The thin strip above x = 2 has zero area, so the area under the curve is the same as A Trapezoid!!!**Discontinuous Integrable Functions**As we already know, a function is not differentiable where it is discontinuous. However, we can integrate functions that have points of discontinuity. Examples… Use areas to show that Sum the rectangles: