Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the following. (If you haven't done Problem 14 , do that first.) (a) \(\int_{-3}^{3}\left|x^{2}-4\right| d x\) (b) \(\int_{0}^{5}|(x+3)(x-1)| d x\)

Step-by-Step Solution

Answer

The value of the first integral \(\int_{-3}^{3}|x^2-4| dx\) is 10/3 and the value of the second integral \(\int_{0}^{5}|(x+3)(x-1)| dx\) is 52/3.

Step 1: Evaluating the first integral

Here, we are integrating \(|x^2 - 4|\) from -3 to 3. To get rid of the absolute value, we need to break this into two integrals based on when \(x^2 - 4\) is negative and when it is positive. This happens when the value of \(x\) is less than -2 and more than 2. Therefore, our integral becomes: \(\int_{-3}^{-2}(-x^2+4) dx + \int_{-2}^{2}(x^2-4) dx + \int_{2}^{3} (-x^2+4) dx\)

Step 2: Compute the value of the first integral

Compute the value of the integrals using the Power Rule for integrating Polynomials. The result is \(2[|(1/3)x^3-4x|]_{-3}^{-2} + 2[(1/3)x^3-4x]_{-2}^{2} + 2[|(1/3)x^3-4x|]_{2}^{3}\)

Step 3: Evaluating the second integral

The second integral involves the absolute value of the product of two linear expressions. We'll proceed in similar manner as above, separating the integral into sectors where the expression inside the absolute value is positive and negative. The zero points of \((x+3)(x-1)\) are -3 and 1, so the integral becomes:\(\int_{0}^{-3}-(x+3)(x-1) dx + \int_{-3}^{1} (x+3)(x-1) dx + \int_{1}^{5} -(x+3)(x-1) dx\)

Step 4: Compute the value of the second integral

Perform integration using the extended power rule for polynomials. This leads to \(2[|1/2x^2+2x|]_{0}^{-3} - 2[1/2x^2+2x]_{-3}^{1} + 2[|1/2x^2+2x|]_{1}^{5}\)

Key Concepts

Absolute Value Functions

Absolute value functions can sometimes make calculus slightly tricky. The absolute value of a function is denoted as \(|f(x)|\), where the value inside depends on whether it is positive or negative. When evaluating integrals involving absolute values, you should split the integral into sections, or intervals, where the expression inside the absolute value maintains a consistent sign. For example, if the expression inside the absolute value is negative over part of the interval and positive in another, you need to compute two separate integrals over those intervals.

Here’s how we handle them:

• First, find the points where the inner expression equals zero.
• These points will act as bounds that determine where the expression changes from positive to negative or vice versa.
• Redefine the function without the absolute value for each interval: change the sign inside the absolute value if it’s negative over that interval.

This method of breaking up the integral allows for smooth integration by avoiding the sudden jump in function values that absolute values create. It's an essential technique for understanding how absolute values can affect the behavior of integrals.

Polynomial Integration

Polynomial integration is an essential part of calculus, and knowing how to integrate polynomials is crucial for evaluating integrals. A polynomial is a mathematical expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

To integrate polynomials, you apply the power rule:

• If you have a term of the form \(x^n\), its antiderivative is \(\frac{x^{n+1}}{n+1}\), where \(n\) is not equal to \(-1\).

This will be repeated for each term of the polynomial. For instance, if you're integrating \(x^2\), its antiderivative is \(\frac{x^3}{3}\).

Moreover, polynomials can be combined with constants, where you integrate them separately, letting you express the result as the sum of those antiderivatives. Always remember to add a constant of integration if you are dealing with indefinite integrals. For definite integrals, evaluate each antiderivative expression with the upper and lower bounds and subtract to find the net area under the curve, ensuring that polynomial functions can be handled with clarity and confidence.

Piecewise Integration

Piecewise integration is a technique used when dealing with functions composed of multiple sub-functions, each valid over a certain interval of the domain. It's especially useful when the function you are integrating consists of different expressions over different ranges. For instance, step functions and functions involving absolute values often require piecewise integration.

The steps include:

• Identify the intervals where each sub-function applies, marking them with their boundaries.
• Integrate each part separately over its designated interval.
• Sum the integrals to cover the entire original function over its entire range of interest.

Considerations must be made at the boundaries as those can cause changes in the function's behavior. Transition points – like where a polynomial changes degree or an absolute value changes sign – need special attention to see how they affect the integral. By applying this method, the integrals remain manageable and ensure accurate results even for complex, piecewise-defined functions.