Prove that the average of LRAM and RRAM is equal to TAM.

Take the definition of the area using the sum of the rectangles using the left endpoint added to the area by using the right endpoints is equal to twice the area using the average midpoint.

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To prove that the average of the Left Riemann Sum (LRAM) and the Right Riemann Sum (RRAM) is equal to the Trapezoidal Approximation Method (TAM), we need to understand each of these concepts.

1. Left Riemann Sum (LRAM): The LRAM is a method used to approximate the area under a curve by dividing the region into rectangles. In this method, the left endpoint of each rectangle is used as the height for that rectangle. The sum of the areas of these rectangles gives an approximation of the area under the curve.

2. Right Riemann Sum (RRAM): The RRAM is similar to the LRAM, but instead of using the left endpoint of each rectangle, it uses the right endpoint as the height for each rectangle.

3. Trapezoidal Approximation Method (TAM): The TAM approximates the area under a curve by dividing the region into trapezoids instead of rectangles. The height of each trapezoid is determined by taking an average of the heights of the left and right endpoints.

Now, let's prove that the average of LRAM and RRAM is equal to TAM.

To analyze this, consider dividing the region into small intervals or subintervals. Let's assume we have n subintervals of equal width Δx.

For the LRAM, the height of each rectangle is the value of the function at the left endpoint of each subinterval. So, the height of the ith rectangle in LRAM is f(x_i), where x_i is the left endpoint of the ith subinterval.

For the RRAM, the height of each rectangle is the value of the function at the right endpoint of each subinterval. So, the height of the ith rectangle in RRAM is f(x_i+Δx), where x_i+Δx is the right endpoint of the ith subinterval.

The area under the curve using LRAM is given by:
LRAM = Δx * [f(x_1) + f(x_2) + ... + f(x_n)]

The area under the curve using RRAM is given by:
RRAM = Δx * [f(x_2) + f(x_3) + ... + f(x_n) + f(x_n+1)]

To find TAM, we take the average of the LRAM and RRAM:
TAM = (LRAM + RRAM) / 2

Substituting the expressions for LRAM and RRAM, we get:
TAM = Δx * [f(x_1) + f(x_2) + ... + f(x_n) + f(x_2) + f(x_3) + ... + f(x_n) + f(x_n+1)] / 2

Combining like terms, we have:
TAM = Δx * [f(x_1) + f(x_2) + 2(f(x_3) + ... + f(x_n)) + f(x_n+1)] / 2

Notice that the summation term in the brackets represents the sum of all the function values except the first and last term, but each term appears twice. Therefore, we can rewrite it as:
TAM = Δx * [f(x_1) + f(x_n+1) + 2(f(x_2) + f(x_3) + ... + f(x_n))] / 2

Further simplifying, we get:
TAM = Δx * [f(x_1) + f(x_n+1) + 2(f(x_2) + f(x_3) + ... + f(x_n))] / 2Δx

The Δx terms cancel out:
TAM = [f(x_1) + f(x_n+1) + 2(f(x_2) + f(x_3) + ... + f(x_n))] / 2

Finally, we can rewrite it as:
TAM = (f(x_1) + 2(f(x_2) + f(x_3) + ... + f(x_n)) + f(x_n+1)) / 2

This is the same expression as the LRAM divided by 2 plus the RRAM divided by 2. Therefore, we can conclude that the average of LRAM and RRAM is equal to TAM.

QED (Quod Erat Demonstrandum), which means "That which was to be proven."