Problem 61 - Calculus Volume 1

Question

In the following exercises, express the limits as integrals. $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5\left(x_{i}^{*}\right)^{2}-3\left(x_{i}^{*}\right)^{3}\right) \Delta x \text { over }[0,2]$$

Step-by-Step Solution

Answer

The limit is expressed as the integral \( \int_{0}^{2} (5x^2 - 3x^3) \; dx \).

Step 1: Understanding the Limit Expression

We are given a limit expression in the form of a Riemann sum, which we need to convert into a definite integral. The expression we have is \( \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5\left(x_{i}^{*}\right)^{2}-3\left(x_{i}^{*}\right)^{3}\right) \Delta x \).

Step 2: Identify the Function and Interval

The function inside the summation is \( f(x) = 5x^2 - 3x^3 \). The interval over which the sum is defined is \([0, 2]\).

Step 3: Rewrite the Sum as an Integral

The sum \( \sum_{i=1}^{n}\left(5\left(x_{i}^{*}\right)^{2}-3\left(x_{i}^{*}\right)^{3}\right) \Delta x \) approximates the integral of \( f(x) = 5x^2 - 3x^3 \) from 0 to 2 as \( n \rightarrow \infty \). Thus, the integral is \( \int_{0}^{2} (5x^2 - 3x^3) \; dx \).

Step 4: Validate the Transition from Limit to Integral

The transition from a Riemann sum to a definite integral is a standard process. The limit of the sum \( \lim _{n \rightarrow \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x \) is equal to the integral \( \int_{a}^{b} f(x) \, dx \) over the interval \([a, b]\). Here, \( a = 0 \) and \( b = 2 \).

Key Concepts

Definite Integral

Definite integrals are a fundamental concept in calculus. They represent the accumulation of quantities over a specific interval on the x-axis. In simpler terms, the definite integral calculates the area under a curve within a given range.

For example, if we have a function, say \( f(x) = 5x^2 - 3x^3 \), the definite integral from 0 to 2 tells us how the values of \( f(x) \) change as x moves from 0 to 2. The integral \( \int_{0}^{2} (5x^2 - 3x^3) \, dx \) helps in understanding this collective change.

• The lower limit (0 in this case) defines where the area starts.
• The upper limit (2 here) defines where the area stops.

Evaluating a definite integral provides a number that expresses the total accumulated value, much like summing up the results of a function over this range.

Riemann Integration

Riemann integration is the technique used to convert discrete sums into continuous integrals. Named after the mathematician Bernhard Riemann, this method splits the area under a curve into many rectangles, calculates the sum of these rectangles, and refines this sum to get an exact area by increasing the number of rectangles, infinitely.

In our exercise, the function \( f(x) = 5x^2 - 3x^3 \) is analyzed from 0 to 2. Imagine dividing this interval into smaller and smaller segments. For each segment, we pick a point, calculate the function value at that point, and multiply it by a tiny length, \( \Delta x \).

• This is done for all segments, creating a sum that approximates the area under the curve.
• As \( n \to \infty \), the sum becomes more accurate and approaches the exact integral \( \int_0^2 (5x^2 - 3x^3) \, dx \).

This method forms the backbone of integral calculus, allowing us to understand the continuous behavior of functions on an interval.

Limit Expression

The concept of limit is crucial in understanding Riemann sums and their transition to integrals. A limit expression evaluates what happens to a function or sum as the variables approach a certain value or infinity.

In the context of Riemann sums, the expression \( \lim _{n \to \infty} \sum_{i=1}^{n} \right) \) involves:

• \( n \) representing the number of sub-intervals.
• Each \( x_i^* \) is a sample point in the sub-interval.
• \( \Delta x \) is the width of each sub-interval.

As \( n \) grows, \( \Delta x \) becomes very small, meaning we have finer partitions of the interval [0,2]. The limit evaluates the behavior and sum of these slices as \( n \to \infty \). Ultimately, this limit is what translates the Riemann sum to a definite integral \( \int_{0}^{2} (5x^2 - 3x^3) \, dx \). This transition sums up the entire area comprehensively and precisely.

Interval Analysis

Interval analysis is a method that focuses on understanding how a function behaves within a specified range. This plays a vital role while working with integrals and Riemann sums, as it dictates the boundaries over which we calculate such sums.

For our problem, the interval is \([0, 2]\). This means we're interested in understanding the behavior of \( f(x) = 5x^2 - 3x^3 \) between x = 0 and x = 2.

• The selection of the interval sets the stage for defining the scope of the integral or Riemann sum.
• Each point in the interval contributes to the eventual calculation of the sum or integral.

This approach allows integrals to not just compute areas but also model and solve real-world problems where understanding a function's behavior in a precise range is crucial, such as physics or engineering scenarios.