Problem 56 - Calculus and its applications

Question

Evaluate. $$ \int_{0}^{3}(x-5)^{2} d x $$

Step-by-Step Solution

Answer

The integral evaluates to 39.

Step 1: Expand the Integrand

First, expand the expression \( (x - 5)^2 \) using the identity \((a-b)^2 = a^2 - 2ab + b^2\). Substitute \(a = x\) and \(b = 5\) to get:\[(x - 5)^2 = x^2 - 2 \cdot x \cdot 5 + 5^2 = x^2 - 10x + 25\]Thus, the integral becomes:\[\int_{0}^{3}(x-5)^{2} dx = \int_{0}^{3}(x^2 - 10x + 25) dx\].

Step 2: Break Down the Integral

Now, integrate each term separately:\[\int_{0}^{3} x^2 \, dx - \int_{0}^{3} 10x \, dx + \int_{0}^{3} 25 \, dx\].

Step 3: Integrate Each Term

Calculate each integral:1. \(\int_{0}^{3} x^2 \, dx\): - The antiderivative of \(x^2\) is \(\frac{x^3}{3}\). - Evaluating from 0 to 3 gives \[\frac{3^3}{3} - \frac{0^3}{3} = 9\].2. \(\int_{0}^{3} 10x \, dx\): - The antiderivative of \(10x\) is \(\frac{10x^2}{2} = 5x^2\). - Evaluating from 0 to 3 gives \[5(3^2) - 5(0^2) = 45\].3. \(\int_{0}^{3} 25 \, dx\): - The antiderivative of 25 is \(25x\). - Evaluating from 0 to 3 gives \[25(3) - 25(0) = 75\].

Step 4: Combine the Results

Add up the results from each integral:\[9 - 45 + 75 = 39\].Thus, the value of the integral is 39.

Key Concepts

Integration Techniques

Integration techniques are strategies or methods used to find the antiderivative or integral of a function. In the given exercise, the focus is on integrating polynomial expressions. There are several useful techniques for integration, including:

• Substitution Method: This technique is useful when dealing with composite functions. By using a change of variables, the integral can be simplified.
• Integration by Parts: Useful for products of functions. It reformulates the integral into a more manageable form.
• Partial Fraction Decomposition: This decomposes a rational function into simpler fractions, easing the integration process.

For polynomial functions like \(x^2 - 10x + 25\), direct integration of each term is straightforward due to their simple structure.However, understanding different techniques can provide you with a toolbox to tackle more complex integrals. Often, polynomials can be decomposed or simplified using algebraic identities, as seen in the expansion step of the exercise. This sets the stage for more precise integration steps.

Polynomial Integration

Polynomial integration involves finding the antiderivative of polynomial expressions. It's one of the simplest integration tasks, thanks to the direct application of the power rule.For any term of the form \(ax^n\), the antiderivative is \(rac{ax^{n+1}}{n+1}\). This rule applies regardless of the polynomial's degree, making integration a routine process.In our exercise, the polynomial \(x^2 - 10x + 25\) is broken down:

• The term \(x^2\) integrates to \(rac{x^3}{3}\).
• The term \(-10x\) integrates to \(-5x^2\).
• The constant term \(25\) integrates to \(25x\).

Once each term's antiderivative is found, they are evaluated over the interval from 0 to 3, as detailed in the solution steps. This evaluation is crucial in determining the definite integral's value, showcasing the straightforward nature of polynomial integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of derivatives and integrals, indicating that they are inverse processes. This theorem consists of two parts:

• The first part tells us that if a function is continuous over an interval, its integral over that interval can be found using any of its antiderivatives.
• The second part shows that the definite integral of a function can be calculated using its antiderivative by evaluating at the boundaries of the interval.

Applying this theorem, the given integral \(\int_0^3 (x^2 - 10x + 25) dx\)\ was evaluated:- Each antiderivative was calculated and evaluated at the limits (0 and 3).- The differences in values give the area under the curve from 0 to 3.This powerful theorem is fundamental in calculus, allowing us to compute exact integral values from antiderivative evaluations, without needing a graphical approach.