Calculating the Absolute Difference: Midpoint Riemann Sum Approximation vs. Exact Integral Value in Math

Let f be the function given by f(x)=x2e−x. It is known that ∫10f(x)ⅆx=0.160603. If a midpoint Riemann sum with two intervals of equal length is used to approximate ∫10f(x)ⅆx, what is the absolute difference between the approximation and ∫10f(x)ⅆx ?

To find the absolute difference between the midpoint Riemann sum approximation and the exact value of the integral, we need to follow these steps:

Step 1: Find the midpoint Riemann sum approximation

To find the absolute difference between the midpoint Riemann sum approximation and the exact value of the integral, we need to follow these steps:

Step 1: Find the midpoint Riemann sum approximation.
Step 2: Calculate the exact value of the integral.
Step 3: Find the absolute difference between the two.

Step 1: Midpoint Riemann Sum Approximation
To approximate the integral using a midpoint Riemann sum with two intervals of equal length, we divide the interval from 1 to 0 into two subintervals of equal length. Let’s denote the width of each interval as Δx. In this case, Δx = (1-0)/2 = 0.5.

Now, we evaluate the function at the midpoint of each subinterval and multiply it by the width Δx. Then, we sum up these values.

f(x) = x^2 * e^(-x)

Approximation using midpoint Riemann sum:
sum = f(0.25 * Δx) + f(0.75 * Δx)
= f(0.25 * 0.5) + f(0.75 * 0.5)
= f(0.125) + f(0.375)

Now, let’s calculate f(0.125) and f(0.375) using the given function:

f(0.125) = (0.125)^2 * e^(-0.125) ≈ 0.015279
f(0.375) = (0.375)^2 * e^(-0.375) ≈ 0.058243

Summing these values:
sum ≈ 0.015279 + 0.058243
≈ 0.073522

Step 2: Exact Value of the Integral
The given integral is:
∫(1 to 0) f(x) dx = 0.160603

Step 3: Absolute Difference
Now, let’s find the absolute difference between the midpoint Riemann sum approximation and the exact value:

Absolute difference = |Approximation – Exact value|
= |0.073522 – 0.160603|
= |-0.087081|
= 0.087081

Therefore, the absolute difference between the midpoint Riemann sum approximation and the exact value of the integral is approximately 0.087081.

More Answers: