Calculating the Area Under a Curve Using Left Riemann Sum: A Step-by-Step Guide

left riemann sum

The left Riemann sum is a method used to approximate the area under a curve by dividing it into several smaller rectangles

The left Riemann sum is a method used to approximate the area under a curve by dividing it into several smaller rectangles. It is a type of Riemann sum that considers the left endpoint of each interval to calculate the height of the rectangle.

To calculate the left Riemann sum, follow these steps:

1. Choose the function you want to approximate the area under and the interval over which you want to calculate the sum.

2. Divide the interval into smaller subintervals of equal width. The width of each subinterval is the difference between the upper and lower limits of the interval divided by the number of subintervals.

3. Determine the left endpoint of each subinterval. For the left Riemann sum, the left endpoint is used as the height of the rectangle.

4. Calculate the area of each rectangle by multiplying the width of the subinterval by the height (the value of the function at the left endpoint).

5. Sum up the areas of all the rectangles to get the approximate total area under the curve.

Here’s a step-by-step example to illustrate the left Riemann sum:

Let’s say we want to approximate the area under the curve of the function f(x) = x^2 from x = 0 to x = 4 using 4 subintervals.

1. We have already chosen the function f(x) = x^2 and the interval from 0 to 4.

2. Divide the interval [0, 4] into 4 subintervals. Each subinterval has a width of (4 – 0) / 4 = 1.

Subintervals: [0, 1], [1, 2], [2, 3], [3, 4]

3. Determine the left endpoints of each subinterval:

Left endpoints: 0, 1, 2, 3

4. Calculate the area of each rectangle:

Rectangle 1: Width = 1, Height = f(0) = 0, Area = 1 * 0 = 0
Rectangle 2: Width = 1, Height = f(1) = 1, Area = 1 * 1 = 1
Rectangle 3: Width = 1, Height = f(2) = 4, Area = 1 * 4 = 4
Rectangle 4: Width = 1, Height = f(3) = 9, Area = 1 * 9 = 9

5. Sum up the areas of all the rectangles:

Total area ≈ 0 + 1 + 4 + 9 = 14

Therefore, the approximate area under the curve of f(x) = x^2 from x = 0 to x = 4 using the left Riemann sum with 4 subintervals is approximately 14 units squared.

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