Chapter 11: Problem 14
Find the indefinite integral and check your result by differentiation. $$ \int 4 y^{-3} d y $$
Chapter 11: Problem 14
Find the indefinite integral and check your result by differentiation. $$ \int 4 y^{-3} d y $$
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Get started for freeUse 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. $$ f(x)=2 x^{2} $$
Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{2}^{5}\left(\frac{1}{x^{2}}-\frac{1}{x^{3}}\right) d x $$
Use integration to find the area of the triangular region having the given vertices. $$ \begin{aligned} &(0,0),(4,0),(4,4) \\ &(0,0),(4,0),(6,4) \end{aligned} $$
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, and find the area of the region. $$ f(x)=x^{2}-4 x, g(x)=0 $$
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