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Chapter 4: Q. 59 (page 353)
Give a geometric argument to prove Theorem 4.13(b): For any real numbers
(Hint: Use a trapezoid.)
The theorem 4.13(b) is proved.
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As n approaches infinity this sequence of partial sums could either converge meaning that the terms eventually approach some finite limit or it could diverge to infinity meaning that the terms eventually grow without bound. which do you think is the case here and why?
Your calculator should be able to approximate the area between a graph and the x-axis. Determine how to do this on your particular calculator, and then, in Exercises 21鈥�26, use the method to approximate the signed area between the graph of each function f and the x-axis on the given interval [a, b].
Prove that in three different ways:
(a) algebraically, by calculating a limit of Riemann sums;
(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;
(c) with formulas, by using properties and formulas of definite integrals.
Fill in each of the blanks:
(a)
(b) is an antiderivative of role="math" localid="1648619282178"
(c) The derivative of is
Consider the sequence A(1), A(2), A(3),.....,A(n) write our the sequence up to n. What do you notice?
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