Animated Solution for Mathematics - Definite Integration: ${\lim }_{n\to \infty }{\sum }_{r=1}^{2n}\frac{r}{\sqrt{n^2+r^2}}$ equals

A Journey into Riemann's World

Welcome, future engineers. Today, we are standing before a problem that looks like a mountain of arithmetic, but is actually a gateway to the elegance of calculus.

We are tasked with evaluating the limit:

At first glance, this looks like a chaotic, impossible addition problem. But I want you to pause. Take a breath. This is not a problem of addition; it is a problem of geometry.

The Riemann Transformation

We are dealing with a classic Riemann Sum. The core philosophy of calculus is to take something discrete—like a collection of thin rectangles—and let the number of those rectangles approach infinity until they become a smooth, continuous curve.

To do this, we need to manipulate our expression into the standard Riemann form:

We need a $\frac{1}{n}$ sitting outside the sum, and we need every $r$ inside the sum to be paired with an $n$ in the denominator.

Algebraic Surgery

Let's look at the denominator: $\sqrt{n^2+r^2}$. It is currently blocking our path. We need to extract an $n$ from this square root.

Watch closely as we perform the surgery:

By factoring out $n^2$, we can pull it out of the square root as $n$. Now, look at the numerator. We have an $r$. If we divide that $r$ by the $n$ we just extracted, we get $\frac{r}{n}$.

This is exactly what we need! Our expression now looks like:

The Calculus Leap

This is the moment of truth. We map our discrete variables to the continuous world. The ratio $\frac{r}{n}$ becomes our continuous variable $x$. The tiny width $\frac{1}{n}$ becomes our infinitesimal differential $dx$.

The summation sign stretches into the integral symbol $\int$. The lower limit is ${\lim }_{n\to \infty }\frac{1}{n}=0$. The upper limit is ${\lim }_{n\to \infty }\frac{2n}{n}=2$.

Our discrete sum has melted into the definite integral:

The Final Cleanup

Now, we solve the integral. We see $x^2$ in the denominator and $x$ in the numerator—a perfect setup for substitution.

Let $t=1+x^2$. Then $dt=2xdx$, or $xdx=\frac{dt}{2}$. We must update our limits: when $x=0$, $t=1$; when $x=2$, $t=5$.

The integral becomes:

Integrating $t^{-1/2}$ gives us $2t^{1/2}$. The $\frac{1}{2}$ and the $2$ cancel out beautifully, leaving us with:

Evaluating this, we get the final result: $\sqrt{5}-1$.

See how the complexity vanished? That is the power of calculus. You didn't just solve a problem; you mastered a transformation.