Estimate the area of the region under the curve y = ln(x) for 1 ≤ x ≤ 4. Use the left hand rule with n = 50. Give your answer to four decimal places.
To estimate the area under the curve y = ln(x) for 1 ≤ x ≤ 4 using the left hand rule with n = 50, we can use the Left Riemann Sum method. This method involves dividing the interval [1, 4] into n subintervals of equal width and using the leftmost point of each subinterval to approximate the height of the curve.
Let's start by finding the width of each subinterval. Since we have n = 50 subintervals and the interval is [1, 4], the width of each subinterval (Δx) can be calculated as:
Δx = (b - a) / n
= (4 - 1) / 50
= 3/50
Next, we need to evaluate the function y = ln(x) at the leftmost point of each subinterval. Using the left hand rule, the x-coordinate of the leftmost point of each subinterval can be calculated as:
xᵢ = a + (i-1) * Δx, where i = 1, 2, 3, ..., n
For n = 50, we have 50 subintervals, so we will evaluate y = ln(x) at x₁, x₂, x₃, ..., x₅₀.
Finally, we compute the sum of the products of the heights (ln(x)) and the widths (Δx):
Riemann Sum = Δx * (y₁ + y₂ + y₃ + ... + y₅₀)
To estimate the area of the region under the curve, we need to calculate this Riemann Sum.
Let's now calculate the estimated area under the curve using the left hand rule with n = 50.
1. Calculate the width of each subinterval:
Δx = (4 - 1) / 50 = 3/50
2. Evaluate the function at the leftmost points of each subinterval:
- Evaluate ln(x) at each xᵢ, where xᵢ = 1 + (i-1) * Δx
- For i = 1 to 50, calculate y(i) = ln(x)
3. Compute the Riemann Sum:
- Riemann Sum = Δx * (y₁ + y₂ + y₃ + ... + y₅₀)
4. Round the final result to four decimal places to obtain the estimated area under the curve.
By following these steps, you can estimate the area under the curve y = ln(x) for 1 ≤ x ≤ 4 using the left hand rule with n = 50.