Problem 2 - Thomas Calculus

Question

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} 2 c_{k}^{3} \Delta x_{k}, \text { where } P \text { is a partition of }[-1,0] $$

Step-by-Step Solution

Answer

\( \int_{-1}^{0} 2x^3 \, dx \)

Step 1: Identify the definite integral form

The given limit expression resembles the definition of a Riemann sum for a definite integral. Recall that a Riemann sum is expressed as \( \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} f(c_k) \Delta x_k \), representing the integral \( \int_{a}^{b} f(x) \, dx \).

Step 2: Determine the function and interval

In the given problem, the integrand function is \( f(x) = 2x^3 \). The partition \( P \) is taken over the interval \([-1,0]\). Therefore, this corresponds to the definite integral of \( 2x^3 \) from \(-1\) to \(0\).

Step 3: Write the definite integral

Using the information from the previous steps, express the limit as a definite integral: \[ \int_{-1}^{0} 2x^3 \, dx \].

Key Concepts

Riemann sum

The concept of a Riemann sum is fundamental in understanding definite integrals. A Riemann sum is a method used to approximate the area under a curve by dividing the region into smaller rectangles or trapezoids.
To construct a Riemann sum:

• Choose a portion of the curve you want to examine, defined by an interval such as \([-1, 0]\) as in this exercise.
• Partition this interval into smaller sub-intervals, each with a width denoted by \(\Delta x_k\).
• Pick a point \(c_k\) within each sub-interval to evaluate the function \(f(c_k)\) at that point. This is often referred to as the sample point.
• Calculate the sum of the areas of the rectangles: \(\sum_{k=1}^{n} f(c_k) \Delta x_k\).

The sum becomes more accurate as the partition gets finer, meaning each \(\Delta x_k\) approaches zero. This process of refining partitions leads us to the concept of the limit, expressed as \(\lim_{\|P\| \rightarrow 0}\). Ultimately, the Riemann sum transitions into a definite integral which provides the exact area under the curve.

integral calculus

Integral calculus is a major branch of mathematics focused largely on accumulation of quantities and the areas under and between curves. It provides tools through which quantities like area, volume, and other physical properties can be calculated.
When dealing with definite integrals, the aim is to find the net area between the curve of a function and the x-axis over a specific interval.

• The definite integral is written as \(\int_{a}^{b} f(x) \, dx\), where \(a\) and \(b\) are the limits of integration.
• The function \(f(x)\) is known as the integrand, and it tells us the shape of the graph under examination.
• The dx signifies that integration is being carried out with respect to \(x\).

For example, in this exercise, the expression \(\int_{-1}^{0} 2x^3 \, dx\) represents the process of finding the exact total area under the curve of \(2x^3\) between the points \(-1\) and \(0\). Definite integrals are particularly vital in physics and engineering, as they allow for precise calculations of properties like displacement, area, and even electricity.

limit expressions

Limit expressions are essential for transitioning from a Riemann sum to a definite integral. A limit helps to determine the value that a function approaches as the input approaches some value.
When computing the limit of a Riemann sum:

• The idea is to let the maximum width of the sub-intervals in partition \(P\) tend towards zero which is expressed as \(\|P\| \rightarrow 0\).
• This refinement leads to a sum that becomes more representative of the actual area under the curve.
• The limit expression then transforms the discrete Riemann sum expression into a continuous integral.

In the exercise, the limit \(\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} 2 c_{k}^{3} \Delta x_{k}\) becomes \(\int_{-1}^{0} 2x^3 \, dx\) as the partition becomes infinitely fine. Essentially, using limits in this context enables calculus to bridge the gap between approximation and precise calculation.