MATH 105 101
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Week 5 Learning Goals
Overview
This week will be about the Fundamental Theorem of Calculus that connects differential
calculus with integration. We use this result to evaluate definite integrals of functions f that
cannot be evaluated with Riemann sums that easily.
We introduce substitution rule and integration by parts as the two most important integration techniques to evaluate definite integrals. For instance, they can be applied to prove
that a circle of radius r has area πr2 .
Parts of this week topics are covered in section 4.8 of the previous textbook, which is
posted online as the scanned copy. Please refer to the online scanned copy for that particular
section.
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Learning objectives
By the end of the week, having participated in lectures, worked through the indicated sections
of the textbook and other resources, and done the suggested problems, you should be able
to:
1. Give the definition of the area function A(x) for a function f (t) with left endpoint a
(Definition p.363).
2. Compute the area function for a given function f (t) using Riemann sums or geometric
interpretation and facts about areas [Procedural].
Example problem: Let f (t) = 2t − 3. Compute the area function A(x) of f (t) with
left endpoint a = −1.
3. Give the definition of an antiderivative of a function f (Definition p.292 §4.8).
4. Give the definition of an indefinite integral of a function f (p.294 §4.8).
5. State the power, constant and sum rule for indefinite integrals (Theorems 4.17 (p.294)
and 4.18 (p.295) §4.8). Write down the indefinite integrals of trigonometric functions such as sin(ax), cos(ax), tan(ax), sec(ax) and apply these to obtain antiderivatives/indefinite integrals for combinations of such functions (Table 4.5 p.296 §4.8) [Procedural].
6. Explain the arbitrary constant involved in the computation of an indefinite integral
(Theorem 4.16 p.293 §4.8). [Conceptual]
7. Explain the distinction between a definite and indefinite integral with the same integrand. [Conceptual]
8. Recognize that the area function A(x) with left end point a of a function f (t) satisfies
A0 (x) = f (x). [Conceptual]
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MATH 105 101
Week 5 Learning Goals
9. State the Fundamental Theorem of Calculus (Theorem 5.3 Part 1 and 2 p.366). [Conceptual]
10. Apply the Fundamental Theorme of Calculus to evaluate definite integrals and compute
derivatives of integrals considered as functions in the lower or upper bound. [Procedural]
Rπ
Example problem: Evaluate the integral 02 cos(x) dx using the Fundamental Theorem of Calculus.
11. State the substitution rule for (in)definite integrals (Theorem 5.6 p.385 and Theorem
5.7 p.388).
12. Apply the substitution rule to transform a given (in)definite integral to a simpler
(in)definite integral. [Procedural]
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Example problem: Evaluate 2 xe2x dx.
13. State the integration by parts rule for (in)definite integrals (p.516 for indefinite integrals
and p.519 for definite integrals).
14. Apply the integration by parts rule to transform a given (in)definite integral into a
simpler (in)definite integral. [Procedural]
R
Example problem: Compute ln x dx.
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