Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{-2}^{4}|x-2| d x$$

Step-by-Step Solution

Answer

The integral evaluates to 10.

Step 1: Understand the Absolute Value Function

The function within the integral is \(|x-2|\). This represents two linear functions depending on whether \(x\) is less than or greater than 2. The absolute value function can be split into two cases: \(x-2\) when \(x \geq 2\) and \(-(x-2)\) when \(x < 2\).

Step 2: Identify Intervals for Integration

Given the limit of integration from \(-2\ to 4\), the point where the function changes is at \(x = 2\). We need to evaluate the integral separately over the intervals \([-2, 2]\) and \([2, 4]\).

Step 3: Integral over the Interval [-2, 2]

For the interval \([-2, 2]\), the integral becomes \(\int_{-2}^{2} -(x-2) \, dx\), which simplifies to \(\int_{-2}^{2} -(x) + 2 \, dx\).

Step 4: Calculate the Integral over [-2, 2]

The integral \(\int_{-2}^{2} -(x) + 2 \, dx\) involves two separate basic integrals: \(-\int_{-2}^{2} x \, dx\) and \(2\int_{-2}^{2} 1 \, dx\). Calculating these:- \(-\int_{-2}^{2} x \, dx = 0 - \left[ \frac{x^2}{2} \right]_{-2}^{2} = 0 - \left((2)^2/2 - (-2)^2/2\right) = 0\)- \(2\int_{-2}^{2} 1 \, dx = 2\left[x\right]_{-2}^{2} = 2(2 - (-2)) = 8\)Adding these results gives \(8\).

Step 5: Integral over the Interval [2, 4]

For the interval \([2, 4]\), the integral is \(\int_{2}^{4} (x-2) \, dx\). Simplify this to \(\int_{2}^{4} x \, dx - 2\int_{2}^{4} 1 \, dx\).

Step 6: Calculate the Integral over [2, 4]

The separate integrals are:- \(\int_{2}^{4} x \, dx = \left[ \frac{x^2}{2} \right]_{2}^{4} = \frac{(4)^2}{2} - \frac{(2)^2}{2} = 8 - 2 = 6\)- \(2\int_{2}^{4} 1 \, dx = 2\left[x\right]_{2}^{4} = 2(4 - 2) = 4\)The total result from these calculations is \(6 - 4 = 2\).

Step 7: Adding Both Areas

Combine the results from intervals \([-2, 2]\) and \([2, 4]\). The value of the integral becomes \(8 + 2 = 10\).

Key Concepts

Absolute Value Function

The absolute value function is crucial in understanding how functions behave on different intervals. Essentially, the absolute value \( |x-2| \) adjusts depending on the value of \( x \), turning any negative result into a positive number. This function is reflected as two linear forms: \( x-2 \) when \( x \geq 2 \) and \(-(x-2)\) or \(-x + 2\) when \( x < 2 \).

In our exercise, understanding this transformation allows us to break down the function into manageable parts, leading to easier integration. Recognizing these linear expressions within the absolute value function ensures accurate interpretations of the areas, making sure calculations are straightforward and efficient.

Area Under a Curve

Finding the area under a curve using integrals reveals a real-world application of calculus. In this specific exercise, our goal is to determine the exact value of the integral \( \int_{-2}^{4} |x-2| \, dx \) by analyzing the areas under each segment of the function.

This approach requires us to consider two separate calculations:

• The area from \(-2\) to \(2\), computed with the expression \(-x + 2\).
• The area from \(2\) to \(4\), using the expression \(x - 2\).

By understanding how to compute these areas separately and summing their results, we obtain the integral's total area. This is analogous to summing the areas of geometric shapes, such as triangles or rectangles, to achieve a complete picture of the space under the curve.

Piecewise Functions

Piecewise functions are functions defined by different equations over different intervals. The integral in our problem, \( \int_{-2}^{4} |x-2| \, dx \), demonstrates the importance of analyzing a function in segments. Here, each piece behaves differently based on the part of the domain it belongs to.

The function \( |x-2| \) splits into:

• From \(-2\) to \(2\), the function \(-x + 2\) applies.
• From \(2\) to \(4\), the function is \(x - 2\).

This division ensures each piece is handled according to its specific rule. Piecewise analysis helps us systematically solve problems involving changing situations within a single function, providing clarity and precision in calculations.