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: Understanding the Problem

We need to express the given limit as a definite integral. The limit given is \( \lim _{\| P | \rightarrow 0} \sum_{k=1}^{n} 2 c_{k}^{3} \Delta x_{k} \). This limit resembles the Riemann sum definition of a definite integral.

Step 2: Identify the Function and Interval

The term \( 2 c_{k}^{3} \) indicates that the function being integrated is \( f(x) = 2x^3 \). The partition \( P \text{ of } [-1,0] \) indicates that the limits of integration are from \(-1\) to \(0\).

Step 3: Write the Definite Integral

Using the Riemann sum definition, we express the limit as the definite integral over the interval \([-1, 0]\) of the function \(f(x) = 2x^3\). Therefore, the integral is \( \int_{-1}^{0} 2x^3 \, dx \).

Key Concepts

Riemann sum

The Riemann sum is a way to approximate the area under a curve, which is vital in understanding the concept of definite integrals. Imagine you have a curve on a graph. The Riemann sum helps you find the area under this curve by breaking it into small sections. Think of it like cutting a pie into many pieces. Each piece represents a small rectangle whose area can be calculated.

To calculate a Riemann sum, you:

• Create a partition of the interval you are interested in. This means breaking the interval into smaller sub-intervals.
• Choose a sample point within each sub-interval. This point gives the height of your rectangle.
• Compute the area for each rectangle by multiplying its height by its width.
• Add up the areas of all the rectangles.

The better your partition and sample points, the closer you get to the actual area under the curve. When these rectangles become extremely thin and numerous, approaching an infinite number, this sum aligns with the concept of the definite integral.

limits of integration

The limits of integration are crucial when dealing with definite integrals. They tell us the interval over which we are summing the Riemann sums to find the area under the curve.

In the exercise, the interval given is from -1 to 0, as indicated by the partition [-1,0]. These limits mean we're calculating the area from the point -1 on the x-axis to the point 0.

Whenever you see an integral like \( \int_{a}^{b} f(x) \, dx \),

• \(a\) is the lower limit of integration. It indicates the starting point on the x-axis for calculation.
• \(b\) is the upper limit of integration. It indicates where the calculation stops on the x-axis.

These limits are essential because they define the region over which you are calculating the area under the curve. Understanding the limits helps you know exactly which portion of the graph you're working with.

integration techniques

Integration techniques are methods used to solve integrals, and they can vary depending on the complexity of the function you are integrating. In the step-by-step solution provided, a straightforward function \(f(x) = 2x^3\) is integrated over an interval.

• The most basic technique is applying the power rule for integration. For functions like \(x^n\), the integral is \(\frac{x^{n+1}}{n+1}\) plus a constant. This is a simple and direct method for polynomial functions.
• Substitution or "u-substitution" is another technique. It's used when you have a composite function, helping simplify the integral into something easier to work with.
• Integration by parts can be effective for products of functions, transforming the integral into a more manageable form.

In our case, the task is made easy by the direct use of the power rule. This is typical for polynomial functions where no complex substitutions or techniques are needed. Understanding which technique to apply depends on the form of the function and can greatly simplify solving the integral.