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 divide the interval [1, 4] into 50 subintervals of equal width.
First, let's calculate the width of each subinterval. We can find the width by dividing the total interval length by the number of subintervals:
Width of each subinterval = (end point - start point) / number of subintervals
= (4 - 1) / 50
= 0.06
Next, we will evaluate the function y = ln(x) at the left endpoint of each subinterval and multiply it by the width to find the area of each rectangle.
For the left-hand rule, we use the x-coordinate of the left endpoint of each subinterval to calculate the corresponding y-coordinate.
In this case, since we have 50 subintervals, the left endpoints will be: 1, 1.06, 1.12, 1.18, 1.24, and so on, until 4.
Now, let's calculate the value of y = ln(x) at each left endpoint:
y₁ = ln(1) ≈ 0
y₂ = ln(1.06)
y₃ = ln(1.12)
and so on, until y₅₀ = ln(4)
Finally, we estimate the area under the curve by summing up the areas of all the rectangles:
Area estimate = (y₁ * width) + (y₂ * width) + (y₃ * width) + ... + (y₅₀ * width)
Evaluating each ln(x) value at the left endpoints and performing the calculations will give us the estimated area.