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Chapter 11: Problem 43
Use a symbolic integration utility to find the indefinite integral. $$ \int y^{2} \sqrt{y} d y $$
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Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x e^{-x^{2}}, y=0, x=0, x=1\\\ &\begin{gathered} 51 / 3 \end{gathered} \end{aligned} $$
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.) $$ f(x)=x\left(x^{2}-3 x+3\right), g(x)=x^{2} $$
Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=3 x^{2}+1 \quad[-1,3] $$
Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{e^{1 / x}}{x^{2}}, y=0, x=1, x=3 $$
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. $$ \frac{d I}{d t}=500 \sqrt{t+1} $$
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