Problem 72 - Calculus

Question

First recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[1+\frac{2 i}{n}+\left(\frac{2 i}{n}\right)^{2}\right] \frac{2}{n} $$

Step-by-Step Solution

Answer

The value of the integral is \(\frac{20}{3}\).

Step 1: Identify the Function

The given sum is a Riemann sum, which can be recognized as an integral. We identify the function inside the summation to be integrated. The expression inside the brackets is: \(1+\frac{2 i}{n}+\left(\frac{2 i}{n}\right)^{2}\).

Step 2: Define the Interval

This sum represents the definite integral over the interval \([a, b]\). Here, the terms \(\frac{2}{n}\) indicate the width of each subinterval, suggesting an interval from \(x = 0\) to \(x = 2\).

Step 3: Express the Sum as an Integral

Convert the sum into an integral with bounds from \(x = 0\) to \(x = 2\). Hence, the limit of the Riemann sum represents the integral \(\int_{0}^{2} \left(1 + x + x^2\right) \, dx\).

Step 4: Apply the Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus allows us to evaluate the integral by finding the antiderivative \(F(x)\) of \(1 + x + x^2\). The antiderivative is \(F(x) = x + \frac{x^2}{2} + \frac{x^3}{3}\).

Step 5: Calculate the Definite Integral

Evaluate the definite integral from 0 to 2 using the antiderivative: \[F(2) - F(0) = \left(2 + \frac{4}{2} + \frac{8}{3}\right) - 0 = 2 + 2 + \frac{8}{3}.\]

Step 6: Compute the Final Result

Combine and simplify the terms: \(2 + 2 + \frac{8}{3} = 4 + \frac{8}{3} = \frac{12}{3} + \frac{8}{3} = \frac{20}{3}\). Therefore, the value of the integral is \(\frac{20}{3}\).

Key Concepts

Riemann sum

A Riemann sum is a way to approximate the total area under a curve on a graph, which we interpret as an integral in calculus. Essentially, it splits the area into a series of rectangles, computes the areas of these rectangles, and then sums them up. By increasing the number of rectangles (making them thinner), the approximation becomes more accurate.

In the expression given in the exercise, \( \lim_{n \rightarrow \infty} \sum_{i=1}^{n}\left[1+\frac{2 i}{n}+\left(\frac{2 i}{n}\right)^{2}\right] \frac{2}{n} \), the sum is a Riemann sum representation of a definite integral. The expression inside the brackets represents the height of each rectangle. The \( \frac{2}{n} \) represents the width of each rectangle. As \( n \) approaches infinity, the rectangles get infinitely thin, producing the exact area under the curve defined by the function \( 1 + x + x^2 \) over the interval from 0 to 2.

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus is a powerful tool connecting differentiation and integration. It allows us to evaluate definite integrals efficiently by focusing on the antiderivative.

• First, find the antiderivative \( F(x) \) of the function you want to integrate.
• Then, evaluate this antiderivative at the bounds of the interval.

In our problem, the function to integrate is \( 1 + x + x^2 \). Its antiderivative, using simple integration rules, is \( F(x) = x + \frac{x^2}{2} + \frac{x^3}{3} \).

To find the definite integral \( \int_{0}^{2} (1+x+x^2) \, dx \), compute \( F(2) - F(0) \). The calculation shows that \( F(2) = 2 + 2 + \frac{8}{3} \) and \( F(0) = 0 \), resulting in a definite integral value of \( \frac{20}{3} \). This theorem transforms the problem of integrating into a simpler problem of subtracting two evaluations of a function.

antiderivative

An antiderivative is a function that reverses the process of differentiation. If you differentiate an antiderivative, you get back your original function. Finding an antiderivative is essential when solving definite integrals using the Second Fundamental Theorem of Calculus.

For the given problem, we needed the antiderivative of the function \( 1 + x + x^2 \). Using integration rules, we found that \( F(x) = x + \frac{x^2}{2} + \frac{x^3}{3} \) is an antiderivative. Here’s how each term is derived:

• The antiderivative of a constant \(1\) is \(x\).
• The antiderivative of \(x\) is \(\frac{x^2}{2}\).
• The antiderivative of \(x^2\) is \(\frac{x^3}{3}\).

Additionally, remember that antiderivatives contain a constant \(+ C\), but for definite integrals, the constant cancels out during evaluation, so we don't need to include it. The role of antiderivatives is crucial in making the connection between differentiation and integration, ultimately allowing us to solve calculus problems more efficiently.