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Chapter 4: Q. 48 (page 404)
Integral Formulas: Fill in the blanks to complete each of the
following integration formulas.
(The last six formulas involve hyperbolic functions and their inverses.)
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Show that is an antiderivative of .
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.
Given a simple proof that
Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.
For each function f and interval [a, b] in Exercises 27–33, 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
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