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Chapter 11: Problem 10
Find the indefinite integral and check your result by differentiation. $$ \int-4 d x $$
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You are given the rate of investment \(d l / d t\). Find the capital accumulation over a five-year period by evaluating the definite integral Capital accumulation \(=\int_{0}^{5} \frac{d l}{d t} d t\) where \(t\) is the time in years. $$ \frac{d I}{d t}=\frac{12,000 t}{\left(t^{2}+2\right)^{2}} $$
Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{1}^{5} \frac{\sqrt{x-1}}{x} d x $$
Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=\frac{1}{x}, \quad[1,5] $$
Use the Trapezoidal Rule with \(n=8\) to approximate the definite integral. Compare the result with the exact value and the approximation obtained with \(n=8\) and the Midpoint Rule. Which approximation technique appears to be better? Let \(f\) be continuous on \([a, b]\) and let \(n\) be the number of equal subintervals (see figure). Then the Trapezoidal Rule for approximating \(\int_{a}^{b} f(x) d x\) is \(\frac{b-a}{2 n}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]\). $$ \int_{0}^{2} x^{3} d x $$
Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ y=\frac{4}{x}, y=x, x=1, x=4 $$
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