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Chapter 4: Q. 45 (page 326)
Find the sum or quantity without completely expanding or calculating any sums.
Given and,. Find the value of.
The sum is.
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Fill in each of the blanks:
(a)
(b) is an antiderivative of .
(c) The derivative of is .
Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.
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.
For each function f and interval [a, b] in Exercises 27鈥�33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.
, n = 3 with
a) Trapezoid sim b) Upper sum
Prove Theorem 4.13(b): For any real numbers a and b, we have. Use the proof of Theorem 4.13(a) as a guide.
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