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Chapter 11: Problem 14
Use the values \(\int_{0}^{5} f(x) d x=6\) and \(\int_{0}^{5} g(x) d x=2\) to evaluate the definite integral. (a) \(\int_{0}^{5} 2 g(x) d x\) (b) \(\int_{5}^{0} f(x) d x\) (c) \(\int_{5}^{5} f(x) d x\) (d) \(\int_{0}^{5}[f(x)-f(x)] d x\)
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Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{2}^{5}\left(\frac{1}{x^{2}}-\frac{1}{x^{3}}\right) d x $$
The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{1}{2} x\right] d x $$
Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(y)=\sqrt{y}, y=9, x=0 $$
Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=e^{x / 4} \quad[0,4] $$
Use the value \(\int_{0}^{1} x^{2} d x=\frac{1}{3}\) to evaluate each definite integral. Explain your reasoning. (a) \(\int_{-1}^{0} x^{2} d x\) (b) \(\int_{-1}^{1} x^{2} d x\) (c) \(\int_{0}^{1}-x^{2} d x\)
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