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Calculus Concept Collection Chapter 5
Antiderivatives and the Indefinite Integral
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Antiderivatives and Differential Equations
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Indefinite Integration: Change of Variable
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1. , use substitution
2. , use substitution
3. , use substitution
4. , use substitution
5. , use substitution
6. , use substitution
7. , use substitution
8. , use substitution
9. , use substitution .
10. , use substitution
11. , use substitution
12. , use substitution
13. , use substitution
14. , use substitution
15. , use substitution
Estimating Area Under a Curve with Finite Riemann Sums
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1. ; (note that we have included areas under the axis as negative values.)
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a. b. c. d.
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a. b. c. d.
11. The graph is symmetric about the origin; hence
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13. The graph is that of a quarter circle of radius ; hence
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The Definite Integral: The Limit of a Riemann Sum
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13. Area=
14. Area=
15. is a parabola with axis of symmetry at , and
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The Definite Integral and the Fundamental Theorem of Calculus
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14. Both functions are odd functions.
a. If the integral in the First Quadrant is from to , then the area enclosed is:
. b. The area enclosed over the First and Third Quadrants is
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Some Basic Properties of Definite Integrals
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14. Apply the Mean Value Theorem for integrals.
15. He is partially correct. The definite integral computes the net area under the curve. However, the area between the curve and the axis is given by
Definite Integrals: The Mean Value Theorem
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11. Average value is
12. By the Mean Value Theorem ; therefore .
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Definite Integration: Change of Variable
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Numerical Integration: Trapezoidal Method
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Numerical Integration: Simpson’s Rule
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13. guarantees an error of no more than .
14. guarantees an error of no more than .
15. guarantees an error of no more than .