Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 7-10, use the given values of a and b and express the given limit as a definite integral. \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\bar{x}_{i}\right)^{3} \Delta x_{i} ; a=1, b=3\)

Step-by-Step Solution

Answer

The definite integral is \(\int_{1}^{3} x^3 \, dx\).

Step 1: Recall the Definition of a Definite Integral

The expression \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n}f(\bar{x}_{i})\Delta x_{i}\) represents the Riemann sum definition of a definite integral. Here, \(f(\bar{x}_{i})\) is the function evaluated at some point \(\bar{x}_{i}\) in the subinterval, and \(\Delta x_{i}\) is the width of the subinterval.

Step 2: Identify the Function f(x)

In the given expression, \(\left(\bar{x}_{i}\right)^3\) is the function \(f(\bar{x}_{i})\). Therefore, the function \(f(x)\) we need to integrate is \(x^3\).

Step 3: Identify the Limits of Integration

The exercise specifies \(a = 1\) and \(b = 3\) as the limits of integration. This means our definite integral will be evaluated from 1 to 3.

Step 4: Formulate the Definite Integral

Using the function \(f(x) = x^3\) from Step 2 and the limits of integration from Step 3, the definite integral expression becomes \( \int_{1}^{3} x^3 \, dx \).

Key Concepts

Riemann Sum

The concept of a Riemann Sum is fundamental in understanding definite integrals. It is a method for approximating the total area under a curve or, in other words, approximating the integral of a function. Imagine dividing the area under a curve into thin rectangles. Each rectangle's width is \(\Delta x_i\), and the height is determined by the function value \(f(\bar{x}_i)\) at a specific point called a sample point within each subinterval.

• A Riemann Sum is expressed as \(\sum_{i=1}^{n} f(\bar{x}_i) \Delta x_i\), where \(n\) is the number of rectangles.
• As the number of rectangles increases (and their width \(\Delta x_i\) becomes smaller), the approximation becomes more accurate.

This eventually leads to the concept of the definite integral as the width of the rectangles approaches zero.

Limit of Integration

When calculating definite integrals, the limits of integration are crucial. They define the interval over which the function is being integrated. The limits are the two numbers \(a\) and \(b\) at the bottom and top of the integral sign, respectively. In a definite integral, you're considering the area under the curve from \(x = a\) to \(x = b\).

• In the example problem, the limits provided are \(a = 1\) and \(b = 3\).
• These limits tell us to evaluate the area under the function from \(x = 1\) to \(x = 3\).

Understanding the limits of integration helps determine the exact portion of the function's curve you are summing up.

Function Evaluation

Function evaluation in the context of definite integrals is determining the value of the function at particular points. In the case of Riemann Sums, this is done within each subinterval of your partition.

• For our problem, the function provided is \(f(x) = x^3\).
• Evaluating \(f\) means replacing \(x\) with values from the interval \([a, b]\) which are used as the sample points \(\bar{x}_i\).

Function evaluation helps in approximating the sum which eventually leads to calculating the definite integral accurately.

Definite Integral Formula

The Definite Integral Formula consolidates all these concepts into a single expression that represents the exact area under a curve over a specified interval.

• The integral \int_{1}^{3} x^3 \, dx\ reflects the total accumulation of the function \(x^3\) from \(1\) to \(3\).
• This integral is computed using the fundamental theorem of calculus, which links differentiable functions and their antiderivatives.

Together, these concepts allow you to express the limit of a Riemann Sum as the exact definite integral, providing a complete and precise area under the curve between the specified limits.