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Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system evaluate the 10-subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{0}^{2}\left(x^{3}+1\right) d x $$

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1}\left(x^{2}+1\right)^{10}(2 x) d x $$

Use the Interval Additive Property and linearity to evaluate \(\int_{0}^{4} f(x) d x .\) Begin by drawing a graph off. $$ f(x)=|x-2| $$

In Problems 35-44, use symmetry to help you evaluate the given integral. $$ \int_{-\pi}^{\pi}(\sin x+\cos x) d x $$

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-1}^{0} \sqrt{x^{3}+1}\left(3 x^{2}\right) d x $$

Use the Interval Additive Property and linearity to evaluate \(\int_{0}^{4} f(x) d x .\) Begin by drawing a graph off. $$ f(x)=3+|x-3| $$

Use symmetry to help you evaluate the given integral. $$ \int_{-1}^{1} \frac{x^{3}}{\left(1+x^{2}\right)^{4}} d x $$

Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system evaluate the 10-subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{0}^{1} \tan x d x $$

Let \(x_{1}, x_{2}, \ldots, x_{n}\) be any real numbers. Find the value of \(c\) that minimizes \(\sum_{i=1}^{n}\left(x_{i}-c\right)^{2}\).

Without doing any calculations, rank from smallest to largest the approximations of \(\int_{1}^{6} \frac{1}{x} d x\) for the following methods: left Riemann sum, midpoint Riemann sum, Trapezoidal rule.