Problem 8 - Calculus

Question

Use the given values of \(a\) and \(b\) and express the given limit as a definite integral. $$ \lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\bar{x}_{i}+1\right)^{3} \Delta x_{i} ; a=0, b=2 $$

Step-by-Step Solution

Answer

The limit is expressed as the integral \(\int_{0}^{2} (x + 1)^3 \, dx\).

Step 1: Identify the Integral Components

Recognize that the limit expression is the definition of a Riemann sum for a definite integral. The function inside the sum is given as \((\bar{x}_i + 1)^3\), and the interval is from \(a = 0\) to \(b = 2\).

Step 2: Set up the Integrand Function

From the Riemann sum expression, identify the integrand function, \(f(x) = (x + 1)^3\). This is the function we will integrate with respect to \(x\).

Step 3: Define the Limits of Integration

The original Riemann sum is taken over the interval \([0, 2]\). Therefore, the limits of integration for the integral are 0 and 2.

Step 4: Write the Definite Integral

Combine the identified integrand and the limits of integration to express the limit as a definite integral. The integral is:\[\int_{0}^{2} (x + 1)^3 \, dx\]

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus that represents the area under a curve, defined by an integrand function, over a specific interval. It serves as a precise way to calculate the accumulation of a quantity, such as area, volume, or other physical quantities.

The process of finding a definite integral of a function involves integrating the function between two limits. The result not only gives a number but also provides insight into how the function behaves across that interval.

To understand how definite integrals work, imagine slicing the area under a curve into numerous vertical strips and summing their areas. As the width of these strips approach zero, the sum becomes exact, and you have a definite integral. This is essentially the process defined by Riemann sums, which forms the basis of definite integration.

Integration Limits

The integration limits define the start and end points of the interval over which the definite integral is calculated. These limits are essential because they determine the part of the graph of the function that you consider when calculating the integral.

When we discuss integration limits, we're looking at two values, often denoted as \(a\) and \(b\), which are the lower and upper limits, respectively. They inform us about the beginning and the end of the region where the area under the curve is calculated.

In the context of definite integrals:

• \(a\) is the lower limit, marking the start of the interval.
• \(b\) is the upper limit, marking the end of the interval.
• The integral of a function \(f(x)\) from \(a\) to \(b\) is written as \(\int_{a}^{b} f(x) \, dx\).

Understanding these limits is critical, especially in applications where the boundaries of the region affect the total value of the accumulated quantity.

Integrand Function

The integrand function is the specific function that you integrate within the process of finding a definite integral. It essentially tells you what function you are calculating the area under.

For instance, in our example where the Riemann sum transitions into a definite integral, the integrand function is \((x + 1)^3\). This function represents the cube of the expression \(x + 1\), and it dictates how the curve behaves over the interval of integration.

The choice of integrand is crucial because:

• It defines the shape of the curve you are integrating.
• Different functions will lead to different areas and therefore different integral values.
• Understanding the behavior of \(f(x)\) is important for correctly setting up and interpreting the integral.

Gaining a strong grasp of the integrand function allows you to accurately perform integrations and interpret the results in practical situations.