Calculating the Area Under a Curve: Understanding and Applying the Left Riemann Sum for Accurate Approximations

Left Reimann Sum

The Left Riemann Sum is a method used in calculus to approximate the area under a curve

The Left Riemann Sum is a method used in calculus to approximate the area under a curve. It is a specific type of Riemann sum, which is the basis of integral calculus.

To understand the Left Riemann Sum, we need to first understand Riemann sums. Imagine you have a curve on a graph and you want to find the area under that curve between two points, say a and b. You can divide the interval [a, b] into n equally sized subintervals. Each subinterval has a width of Δx = (b – a) / n. By finding the area of each rectangle with a base on a subinterval and a height equal to the value of the curve at the left endpoint of that subinterval, you can sum up the areas of all the rectangles to get an approximation of the area under the curve.

Now, specifically for the Left Riemann sum, the height of each rectangle is equal to the value of the curve at the left endpoint of the subinterval. This means that for each subinterval [x₀, x₁], the height of the rectangle will be f(x₀), where f(x) is the function representing the curve.

To calculate the Left Riemann Sum, you follow these steps:

1. Divide the interval [a, b] into n equally sized subintervals. The width of each subinterval is Δx = (b – a) / n.
2. Identify the left endpoint of each subinterval. The left endpoint of the first subinterval is a, and the left endpoint of the i-th subinterval is xᵢ = a + (i-1) * Δx.
3. Evaluate the function at each left endpoint. Calculate f(xᵢ) for each i=1,2,…,n.
4. Multiply each f(xᵢ) by the width Δx to get the area of each rectangle.
5. Sum up all the areas of the rectangles to get the Left Riemann Sum, which is an approximation of the area under the curve.

Mathematically, the Left Riemann Sum can be expressed as:

L_n = Δx * (f(x₀) + f(x₁) + f(x₂) + … + f(xₙ₋₁))

where L_n is the Left Riemann Sum, f(x) is the function representing the curve, and x₀, x₁, …, xₙ₋₁ are the left endpoints of each subinterval in the interval [a, b].

It is important to note that as the number of subintervals (n) approaches infinity, the Left Riemann Sum becomes a precise approximation of the area under the curve, which is known as the definite integral.

The Left Riemann Sum, along with other Riemann sums, plays a key role in understanding and calculating definite integrals and the area under curves in calculus.

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