1. Homework Homework Assignment #2 Read Section 5.3
Page 321, Exercises: 1 – 61(EOO), 71, skip 41
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2. Example, Page 321
Draw a graph of the signed area represented by the integral and compute it using geometry.
Rogawski Calculus
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3. Example, Page 321
Draw a graph of the signed area represented by the integral and compute it using geometry.
Rogawski Calculus
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4. Example, Page 321
Draw a graph of the signed area represented by the integral and compute it using geometry.
Rogawski Calculus
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5. Example, Page 321
13. Evaluate the integrals for f (x) shown in Figure 14.
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6. Example, Page 321
Sketch the signed area represented by the integral. Indicate the regions of positive and negative area.
Rogawski Calculus
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7. Example, Page 321
Sketch the signed area represented by the integral. Indicate the regions of positive and negative area.
Rogawski Calculus
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8. Example, Page 321
Determine the sign of the integral without calculating it. Draw a graph if necessary.
The sign of the integral
is negative.
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9. Example, Page 321
Calculate the Riemann sum for the given function, partition, and choice of intermediate points. Also, sketch the graph of f and the rectangles corresponding to R (f, P, C).
Rogawski Calculus
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10. Example, Page 321
Use the basic properties of the integral and the formulas in the summary to calculate the integral.
Rogawski Calculus
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11. Example, Page 321
Use the basic properties of the integral and the formulas in the summary to calculate the integral.
Rogawski Calculus
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12. Example, Page 321
Use the formulas in the summary and Equation 9 to evaluate the integral.
Rogawski Calculus
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13. Example, Page 321
Use the formulas in the summary and Equation 9 to evaluate the integral.
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14. Example, Page 321 Calculate the integral, assuming Rogawski Calculus
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15. Example, Page 321 Calculate the integral, assuming Rogawski Calculus
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16. Example, Page 321 Express each integral as a single integral.
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17. Example, Page 321
Calculate the integral, assuming f is an integrable function such .
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18. Example, Page 321 Calculate the integral. Rogawski Calculus
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19. Calculus, ET First Edition
Jon Rogawski
Calculus, ET First Edition
Chapter 5: The Integral
Section 5.3: The Fundamental Theorem of Calculus, Part I
Rogawski Calculus
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20. The Fundamental Theorem of Calculus ties together the operations of
differentiation and integration in a relationship. Specifically, the
integral of f (x) evaluated from a to b is equal to the antiderivative of
f (x) evaluated at x = b minus the antiderivative of f (x) evaluated at
x = a.
Rogawski Calculus
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21. Verify the assertion about the area under the curve in Figure 2.
Rogawski Calculus
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22. Find the area under the curve between the limits shown in figure 3.
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23. Verify that the shaded are in Figure 4 is 2. Rogawski Calculus
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24. Calculate the shaded area in Figure 5.
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25. Recalling that the antiderivative of f (x) = x–1 is F(x) = ln x, the FTC
tells us:
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26. Verify the relationship in the caption to Figure 6.
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27. Find the areas of the shaded regions in Figure 7. Rogawski Calculus
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28. Basic Antiderivative Formulae
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29. Example, Page 329 Evaluate the integral using the FTC 1.
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30. Example, Page 329 Evaluate the integral using the FTC 1.
Rogawski Calculus
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31. Example, Page 329 Evaluate the integral using the FTC 1.
Rogawski Calculus
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32. Example, Page 329 Evaluate the integral using the FTC 1.
Rogawski Calculus
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33. Example, Page 329 Evaluate the integral using the FTC 1.
Rogawski Calculus
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34. Example, Page 329
Write the integral as a sum of integrals without the absolute values and evaluate.
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35. Example, Page 329 Evaluate the integral in terms of the constants.
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36. Homework Homework Assignment #3 Read Section 5.4
Page 329, Exercises: 1 – 49(EOO)
Quiz next time
Rogawski Calculus
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