Estimate $I=\int_{5}^{9}\left(\frac{4}{x}+1\right) d x$ using $n=4$ subintervals and (a) Left endpoints. \[ I \approx \] (b) Right endpoints. \[ I \approx \]
Solution
Step 1 ：Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n}\), where \(a=5\), \(b=9\), and \(n=4\). This gives \(\Delta x = 1.0\).
Step 2 ：Calculate the \(x\)-values of the left endpoints of the subintervals. These are \(x=a, a+\Delta x, a+2\Delta x, \ldots, a+(n-1)\Delta x\), which gives \(x = [5, 6, 7, 8]\).
Step 3 ：Evaluate the function \(f(x) = \frac{4}{x} + 1\) at these \(x\)-values. This gives \(f(x) = [1.8, 1.66666667, 1.57142857, 1.5]\).
Step 4 ：Calculate the estimated integral using the formula \(\Delta x \sum_{i=1}^{n} f(x_i)\), where the \(x_i\) are the left endpoints. This gives \(I \approx 6.54\).
Step 5 ：Calculate the \(x\)-values of the right endpoints of the subintervals. These are \(x=a+\Delta x, a+2\Delta x, \ldots, a+n\Delta x\), which gives \(x = [6, 7, 8, 9]\).
Step 6 ：Evaluate the function \(f(x) = \frac{4}{x} + 1\) at these \(x\)-values. This gives \(f(x) = [1.66666667, 1.57142857, 1.5, 1.44444444]\).
Step 7 ：Calculate the estimated integral using the formula \(\Delta x \sum_{i=1}^{n} f(x_i)\), where the \(x_i\) are the right endpoints. This gives \(I \approx 6.18\).
Step 8 ：The estimated integral using the left endpoints is approximately 6.54, and the estimated integral using the right endpoints is approximately 6.18. These are estimates of the exact value of the integral, and the actual value will be somewhere between these two estimates.
Step 9 ：\(\boxed{\text{Final Answer: For the left endpoints, } I \approx 6.54 \text{. For the right endpoints, } I \approx 6.18}\)
