Chapter 5: Problem 20
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { \cos x } ^ { \pi / 2 } \cos x d x$$
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In Exercises \(23-26\) use a calculator program to find the Simpson's Rule approximations with \(n=50\) and \(n=100 .\) $$\int_{0}^{\pi / 2} \frac{\sin x}{x} d x$$
In Exercises \(23-26\) use a calculator program to find the Simpson's Rule approximations with \(n=50\) and \(n=100 .\) $$\int_{0}^{1} \sqrt{1+x^{4}} d x$$
Multiple Choice If \(\int _ { 2 } ^ { 5 } f ( x ) d x = 12\) and \(\int _ { 5 } ^ { 8 } f ( x ) d x = 4\) then all of the following must be true except (A) $$\int _ { 2 } ^ { 8 } f ( x ) d x = 16$$ (B) $$\int _ { 2 } ^ { 5 } f ( x ) d x - \int _ { 5 } ^ { 8 } 3 f ( x ) d x = 0$$ (C) $$\int _ { 5 } ^ { 2 } f ( x ) d x = - 12$$ (D) $$\int _ { - 5 } ^ { - 8 } f ( x ) d x = - 4$$ (E) $$\int _ { 2 } ^ { 6 } f ( x ) d x + \int _ { 6 } ^ { 8 } f ( x ) d x = 16$$
Revenue from Marginal Revenue Suppose that a company's marginal revenue from the manufacture and sale of egg beaters is \(\frac{d r}{d x}=2-\frac{2}{(x+1)^{2}}\)where \(r\) is measured in thousands of dollars and \(x\) in thousands of units. How much money should the company expect from a production run of \(x=3\) thousand eggbeaters? To find out, integrate the marginal revenue from \(x=0\) to $x=3 . \quad \$
Consider the integral \(\int_{0}^{\pi} \sin x d x\) (a) Use a calculator program to find the Trapezoidal Rule approximations for n = 10, 100, and 1000. (b) Record the errors with as many decimal places of accuracy as you can. (c) What pattern do you see? (d) Writing to Learn Explain how the error bound for \(E_{T}\) accounts for the pattern.
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