Estimate the area of the region under the curve y = ln(x) for 1 ≤ x ≤ 5. Use the left-hand rule with n = 50. (Round your answer to four decimal places.)
To estimate the area under the curve y = ln(x) for 1 ≤ x ≤ 5 using the left-hand rule with n = 50, follow these steps:
1. Divide the interval [1, 5] into n subintervals of equal width. Since n = 50, each subinterval will have a width of (5 - 1) / 50 = 0.08.
2. Determine the x-coordinate for the left endpoint of each subinterval. Start with x = 1 and increment it by 0.08 for each subsequent subinterval. This will give you the x-coordinates 1, 1.08, 1.16, 1.24, and so on, until you reach 5.
3. Evaluate the function y = ln(x) at each of these x-coordinates to find the corresponding y-values. Calculate ln(x) for each x-value you obtained in step 2. This will give you the y-values ln(1), ln(1.08), ln(1.16), ln(1.24), and so on.
4. Multiply each y-value by the width of the subinterval (0.08) to find the area of the rectangle formed under each subinterval. This represents the approximation of the area between the curve and the x-axis for each subinterval.
5. Sum up all the areas of these rectangles to get the estimated area under the curve. Add up the areas calculated in step 4 for each subinterval to get the total estimated area.
Following these steps will allow you to estimate the area of the region under the curve y = ln(x) for 1 ≤ x ≤ 5 using the left-hand rule with n = 50.