Understanding the Concept of "Limit as a Sum"

The concept of "limit as a sum" is fundamental in calculus and is closely related to the definition of definite integrals. It is a method of finding the area under a curve by approximating it with the sum of areas of rectangles. This method is also known as Riemann sums. Let's delve deeper into this concept.

Definite Integrals as Limits of Sums

A definite integral of a function over an interval [a, b] can be interpreted as the limit of a sum of areas of rectangles as the number of rectangles approaches infinity. Mathematically, this is expressed as:

$$ \int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x $$

where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is a sample point in the $i$-th subinterval.

Table: Key Differences and Important Points

Aspect	Riemann Sum	Definite Integral
Definition	Approximation of the area under a curve using a finite number of rectangles.	The limit of Riemann sums as the number of rectangles approaches infinity.
Expression	$\sum_{i=1}^{n} f(x_i^*) \Delta x$	$\int_{a}^{b} f(x) \, dx$
$\Delta x$	Width of each rectangle.	Tends to 0 as $n \to \infty$.
$x_i^*$	Any point in the $i$-th subinterval.	Becomes irrelevant as $n \to \infty$.
Interval Partition	Finite number of subintervals.	Infinite number of points in the interval.
Accuracy	Depends on the number of rectangles and the function's behavior.	Exact area under the curve.

Formulas and Examples

Riemann Sums

There are different types of Riemann sums based on the choice of $x_i^*$:

1. Left Riemann Sum: $x_i^* = a + (i-1)\Delta x$
2. Right Riemann Sum: $x_i^* = a + i\Delta x$
3. Midpoint Riemann Sum: $x_i^* = a + \left(i-\frac{1}{2}\right)\Delta x$

Example 1: Left Riemann Sum

Let's approximate the area under the curve $f(x) = x^2$ over the interval [0, 2] using a Left Riemann Sum with $n = 4$ subintervals.

• $\Delta x = \frac{2-0}{4} = 0.5$
• $x_i^* = 0 + (i-1)\Delta x = 0.5(i-1)$

The sum is:

$$ \sum_{i=1}^{4} f(x_i^*) \Delta x = \sum_{i=1}^{4} (0.5(i-1))^2 \cdot 0.5 $$

Calculating the sum gives us an approximation of the area under the curve.

Example 2: Definite Integral as a Limit of a Sum

Now, let's find the exact area under the same curve $f(x) = x^2$ over the interval [0, 2] using the limit of a sum.

$$ \int_{0}^{2} x^2 \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \left(\frac{2i}{n}\right)^2 \cdot \frac{2}{n} $$

This limit can be computed using the properties of limits and sums to yield the exact value of the integral, which is $\frac{8}{3}$.

Conclusion

The "limit as a sum" approach is a powerful technique in calculus for finding the area under a curve. It bridges the gap between finite sums and the concept of integration. By understanding and applying this method, one can solve a wide range of problems involving areas and accumulate changes in various fields of science and engineering.