Estimate the area under the graph of $f(x)=\frac{1}{x^{2}+1}$ over the interval $[1,4]$ using eight approximating rectangles and right endpoints. \[ R_{n}= \] Repeat the approximation using left endpoints. \[ L_{n}= \] Report answers accurate to 4 places. Remember not to round too early in your calculations. Question Help: $\square$ Video $\square$ Message instructor Submit Question Jump to Answer
Solution
Step 1 ：The problem is asking for the estimation of the area under the graph of the function \(f(x)=\frac{1}{x^{2}+1}\) over the interval \([1,4]\) using eight approximating rectangles and right endpoints. This is a typical problem of numerical integration using the method of Riemann sums.
Step 2 ：The Riemann sum \(R_n\) using right endpoints is given by the formula: \[R_{n}=\sum_{i=1}^{n}f(x_{i}^{*})\Delta x\] where \(x_{i}^{*}\) is the right endpoint of the i-th subinterval and \(\Delta x\) is the width of each subinterval.
Step 3 ：Similarly, the Riemann sum \(L_n\) using left endpoints is given by the formula: \[L_{n}=\sum_{i=1}^{n}f(x_{i}^{*})\Delta x\] where \(x_{i}^{*}\) is the left endpoint of the i-th subinterval.
Step 4 ：In this case, the interval \([1,4]\) is divided into 8 equal subintervals, so \(\Delta x=\frac{4-1}{8}=\frac{3}{8}\).
Step 5 ：The right endpoints are \(x_{i}^{*}=1+i\Delta x\) for \(i=1,2,...,8\) and the left endpoints are \(x_{i}^{*}=1+(i-1)\Delta x\) for \(i=1,2,...,8\).
Step 6 ：We can calculate \(R_n\) and \(L_n\) by substituting these values into the formulas and using the given function \(f(x)=\frac{1}{x^{2}+1}\).
Step 7 ：The estimated area under the graph of \(f(x)=\frac{1}{x^{2}+1}\) over the interval \([1,4]\) using eight approximating rectangles and right endpoints is \(R_{n}=\boxed{0.4632}\) and using left endpoints is \(L_{n}=\boxed{0.6287}\).
