91Ó°ÊÓ

Use the tabulated values of \(f\) to evaluate the left and right Riemann sums for the given value of \(n\). $$n=4 ;[0,2]$$ $$\begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 0.5 & 1 & 1.5 & 2 \\ \hline f(x) & 5 & 3 & 2 & 1 & 1 \\ \hline \end{array}$$

Mean Value Theorem for Integrals Find or approximate the point\((s)\) at which the given function equals its average value on the given interval. $$f(x)=8-2 x ;[0,4]$$

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{-2}^{2}\left(x^{2}-4\right) d x$$

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\ln 8} e^{x} d x$$

Variations on the substitution method Find the following integrals. $$\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$$

Use the tabulated values of \(f\) to evaluate the left and right Riemann sums for the given value of \(n\). $$n=8 ;[1,5]$$ $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline x & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 & 5 \\ \hline f(x) & 0 & 2 & 3 & 2 & 2 & 1 & 0 & 2 & 3 \\ \hline \end{array}$$

Mean Value Theorem for Integrals Find or approximate the point\((s)\) at which the given function equals its average value on the given interval. $$f(x)=e^{x} ;[0,2]$$

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{1 / 2}^{1}\left(x^{-3}-8\right) d x$$

Variations on the substitution method Find the following integrals. $$\int x \sqrt[3]{2 x+1} d x$$

The accompanying figure shows four regions bounded by the graph of \(y=x \sin x: R_{1}, R_{2}, R_{3},\) and \(\mathrm{R}_{4}\) whose areas are \(1, \pi-1, \pi+1,\) and \(2 \pi-1,\) respectively. Use this information to evaluate the following integrals. $$\int_{0}^{\pi} x \sin x d x$$