91Ó°ÊÓ

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$R_{n}=\frac{3}{n} \sum_{i=1}^{n}\left(3+3 \frac{i}{n}\right)$$

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$L_{n}=\frac{2 \pi}{n} \sum_{i=1}^{n} 2 \pi \frac{i-1}{n} \cos \left(2 \pi \frac{i-1}{n}\right)$$

In the following exercises, given \(L_{n}\) or \(R_{n}\) as indicated,express their limits as \(n \rightarrow \infty\) as definite integrals, identifying the correct intervals. $$R_{n}=\frac{1}{n} \sum_{i=1}^{n}\left(1+\frac{i}{n}\right) \log \left(\left(1+\frac{i}{n}\right)^{2}\right)$$

In the following exercises, evaluate the integral using area formulas. \(\int_{0}^{3}(3-x) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{2}^{3}(3-x) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{-3}^{3}(3-|x|) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{0}^{6}(3-|x-3|) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{-3}^{3}(3-|x|) d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{1}^{5} \sqrt{4-(x-3)^{2}} d x\)

In the following exercises, evaluate the integral using area formulas. \(\int_{0}^{12} \sqrt{36-(x-6)^{2}} d x\)