Problem 1 - Essential Calculus

Question

In Exercises 1 and 2, use Example 1 as a model to evaluate the limit $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(c_{i}\right) \Delta x_{i}$$ over the region bounded by the graphs of the equations. $$ \begin{array}{l} f(x)=\sqrt{x}, \quad y=0, \quad x=0, \quad x=3 \\ \left(\text { Hint: } \text { Let } c_{i}=3 i^{2} / n^{2} .\right) \end{array} $$

Step-by-Step Solution

Answer

The limit of the Riemann Sum for the given function and region is 2.

Step 1: Break down the region into subintervals

Divide the interval from 0 to 3 into n even subintervals, each of width \(\Delta x = \frac{3}{n}\). The ith subinterval goes from x = \((i-1) \Delta x\) to \(i \Delta x\).

Step 2: Use the provided hint

Here, \(c_i = 3 i^2/ n^{2}\) is taken to be in the ith subinterval. Now, find the altitude of the rectangle on this subinterval, which is \(f(c_i)\). So, substitute \(f(x)=\sqrt{x}\) with \(c_i\) into the function \(f(x)\) to get the height \(f(c_i) = \sqrt{c_i} = \sqrt{3i^{2}/n^{2}}\).

Step 3: Express the sum as a limit of function

The sum \( \sum_{i=1}^{n} f\left(c_{i}\right) \Delta x_{i}\) evaluates to \( \sum_{i=1}^{n} \sqrt{3i^{2}/n^{2}} \cdot \frac{3}{n}\). Now, apply the limit as \(n \rightarrow \infty\).

Step 4: Evaluate the limit

As the number of rectangles increases, the sum becomes the exact total area under the curve, which is \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \sqrt{3i^{2}/n^{2}} \cdot \frac{3}{n}\). After computing, the result is 2. The final answer is thus 2.

Key Concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It's a powerful tool for analyzing change and motion, which has wide applications in science, engineering, economics, and beyond. At the heart of calculus is the concept of a limit, a fundamental notion which describes the behavior of a function as its argument either gets close to some point, or as it grows without bound.

For instance, when we talk about evaluating the limit as in the textbook exercise, we are essentially interested in knowing what value the sum of the function's outputs approaches as the number of subintervals increases indefinitely. This process allows us to ponder on what happens to the summation of infinitely many small quantities, which is how we connect calculus with concepts like the Riemann sums and infinite series.

Riemann Sum

The Riemann Sum is a method of approximating the total area under a curve on a graph, which represents a function. This is done by dividing the area into a number of rectangular strips, calculating the area of each one, and then summing up these areas. The more strips used, the better the approximation becomes. The width of each strip is typically denoted as \( \Delta x \), and the height is the function value at a chosen point within the subinterval, \( f(c_i) \).

In relation to our exercise, we used the provided hint to set \( c_i = \frac{3i^2}{n^2} \) which lies within the ith subinterval. The area of each individual rectangle becomes the product of the function value at \( c_i \) and the width of the strip \( \Delta x_i \). As \( n \), the number of rectangles, approaches infinity, these sums are called the Riemann sums of the function, and they approach the precise area under the curve, which can be represented by a definite integral.

Infinite Series

An infinite series is the summation of an infinite sequence of numbers, which can be thought of as an ongoing addition that extends indefinitely. The connection between infinite series and calculus exists in their mutual exploration of infinitesimally small quantities and how they sum up to define a whole. This is precisely what we investigate when we evaluate the limit of the Riemann sum as the number of subintervals \( n \) goes to infinity.

In the given example, when we take the limit as \( n \to \infty \), we're calculating the infinite series formed by the areas of the rectangles under the curve \( f(x) = \sqrt{x} \). This allows us to precisely estimate the integral of the function over the interval \[0,3\]. With this understanding and careful calculation, we determined that the infinite series, or the precise area under the given curve, is 2.