91Ó°ÊÓ

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. $$\sum_{k=1}^{25}\left[(2 k)^{2}-100 k\right]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$L_{4} \text { for } f(x)=\frac{1}{x-1} \text { on }[2,3]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$R_{4} \text { for } g(x)=\cos (\pi x) \text { on }[0,1]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$L_{6} \text { for } f(x)=\frac{1}{x(x-1)} \text { on }[2,5]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$R_{6} \text { for } f(x)=\frac{1}{x(x-1)} \text { on }[2,5]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$L_{4} \text { for } \frac{1}{x^{2}+1} \text { on }[-2,2]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$R_{4} \text { for } x^{2}-2 x+1 \text { on }[0,2]$$

Let \(L_{n}\) denote the left-endpoint sum using \(n\) subintervals and let \(R_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval. $$L_{8} \text { for } x^{2}-2 x+1 \text { on }[0,2]$$

Compute the left and right Riemann sums \(-L_{4}\) and \(R_{4}\) , respectively- for \(f(x)=(2-|x|)\) on \([-2,2] .\) Compute their average value and compare it with the area under the graph of \(f .\)

Compute the left and right Riemann sums- \(L_{6}\) and \(R_{6},\) respectively-for \(f(x)=(3-|3-x|)\) on [0,6] Compute their average value and compare it with the area under the graph of \(f\).