Problem 20 - Applied Calculus

Question

Use a calculator or computer to evaluate the integral. $$ \int_{0}^{5} x^{2} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{125}{3} \).

Step 1: Understand the Problem

We need to evaluate the definite integral \( \int_{0}^{5} x^{2} \mathrm{d} x \). This represents the area under the curve of \( y = x^2 \) from \( x = 0 \) to \( x = 5 \).

Step 2: Set Up the Integral

The integral to calculate is \( \int_{0}^{5} x^{2} \mathrm{d} x \). This integral requires finding the antiderivative of \( x^2 \) and then evaluating it from 0 to 5.

Step 3: Find the Antiderivative

The antiderivative of \( x^2 \) is \( \frac{x^3}{3} \), which is derived from the power rule of integration.

Step 4: Evaluate the Antiderivative at the Boundaries

Plug in the upper limit (5) and the lower limit (0) into the antiderivative: \[ \left[ \frac{x^3}{3} \right]_{0}^{5} = \frac{5^3}{3} - \frac{0^3}{3} \].

Step 5: Calculate the Result

Calculate \( \frac{5^3}{3} - \frac{0}{3} = \frac{125}{3} \). Thus, the value of the integral is \( \frac{125}{3} \).

Key Concepts

Power Rule of Integration

The power rule of integration is a fundamental concept in calculus, which helps us find the antiderivative of polynomial functions. If you're familiar with the power rule for derivatives, this is essentially its reverse. In the power rule of integration, if you have a function of the form \( f(x) = x^n \), where \( n eq -1 \), its antiderivative is given by:

• \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

Here, \( C \) is an arbitrary constant that appears in indefinite integrals but not in definite integrals.
Definite integrals, which have specified limits, compare values at two boundaries, as we will see in our example.In the context of our exercise, remember that the integral \( \int_{0}^{5} x^2 \mathrm{d}x \) applies the power rule. We increase the exponent from 2 to 3 and divide by the new exponent, resulting in the antiderivative \( \frac{x^3}{3} \). This is a core part of solving integrals analytically, especially for polynomial functions.

Antiderivative

An antiderivative, also known as an integral or primitive, is a function whose derivative matches the original function. In simpler terms, it's about finding a function that reverses the process of differentiation.
This is essential in problems involving indefinite integration or when calculating area under curves in definite integrals.In the example given, our goal was to find an antiderivative for \( x^2 \). By using the power rule, we determined that this is \( \frac{x^3}{3} \).
Remember:

• Finding an antiderivative is like finding out what function differentiated gives you the initial function.
• Understanding the transition from differentiation (tangent slopes) to integration (area under a curve) is fundamental in calculus.

Antiderivatives help in a variety of applications, from physics to engineering, where evaluating changes and areas are necessary. Recognizing the antiderivative step enables us to proceed with evaluating definite integrals.

Calculus Problem Solving

Calculus problem solving often involves a step-by-step approach to integrate mathematical concepts seamlessly. In the exercise, we tackled a calculus problem involving a definite integral.
Here's how you can think of similar tasks:

• **Understand the Problem:** Identify that you need to find the area under a curve between two limits. Ask yourself what the integral represents.
• **Set Up the Integral:** Confirm the function you’ll integrate and the limits of integration.
• **Find the Antiderivative:** Use the appropriate rule of integration, like the power rule, to find the antiderivative needed.
• **Evaluate at Boundaries:** In definite integrals, plug the limits into the antiderivative to find the result.
• **Calculate the Result:** Perform necessary arithmetic to find the exact value.

By optimizing these analysis steps and applying calculus principles like the power rule of integration and antiderivatives, one can solve complex problems efficiently and confidently. Mastery of these concepts expands your mathematical toolkit, allowing you to dissect problems into manageable steps every time!