Question

What is the whole process of turning the limit into a definite integral? An example would be great!

Answer 1

Step-by-step explanation:

The first step is to find the (b − a) / n factor in the limit of the Riemann sum.  This is the Δx.

The next step is to identify the function and the argument, f(xᵢ).  The argument xᵢ looks like a + i (b − a) / n.

From there, you can write two equations:

Δx = (b − a) / n

xᵢ = a + i (b − a) / n = a + i Δx

Since you know xᵢ and Δx, you can identify a.  And when you know a, you can plug that into Δx to find b.  So now you have the limits of the definite integral.

Finally, write the integral using the limits and the function.

Here's an example.  Suppose you have the limit:

lim(n→∞) ∑ᵢ₌₁ⁿ √(1 + 3i/n) (3/n)

First, we notice that Δx = (b − a)/n = 3/n.  So b − a = 3.

Next, we identify that xᵢ = 1 + 3i/n = a + i (3/n), so a = 1.  Therefore, b = 4.

Now, we identify the function, f(xᵢ) = √xᵢ.

Finally, we plug it all into a definite integral:

∫₁⁴ √x dx