{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of ƒ and the 𝓍-axis on the given interval. Graph ƒ and show the region 𝑅.                                              
                                                                                                                                                                                    
 ƒ(𝓍) = 2 ― |𝓍| on [ ― 2 , 4]

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the area of the region R bounded by the graph of ƒ(x) = 2 - |x| and the x-axis over the interval [-2, 4]. This involves integrating the function ƒ(x) over the given interval.
2. Step 2: Break the function ƒ(x) = 2 - |x| into piecewise components because the absolute value function |x| behaves differently for x < 0 and x ≥ 0. For x < 0, |x| = -x, and for x ≥ 0, |x| = x. Rewrite ƒ(x) as: ƒ(x) = 2 - (-x) = 2 + x for x in [-2, 0], and ƒ(x) = 2 - x for x in [0, 4].
3. Step 3: Set up the integral to calculate the area. Since the function changes at x = 0, split the integral into two parts: ∫[−2,0](2 + x) dx + ∫[0,4](2 − x) dx.
4. Step 4: Compute each integral separately. For the first integral ∫[−2,0](2 + x) dx, use the power rule and evaluate the definite integral. For the second integral ∫[0,4](2 − x) dx, similarly apply the power rule and evaluate the definite integral.
5. Step 5: Add the results of the two integrals to find the total area of the region R. Ensure that any negative values from the integrals are treated as positive since area is always non-negative.

Key Concepts

Definite Integral

The definite integral of a function over a specified interval represents the net area between the graph of the function and the x-axis. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The area can be positive or negative depending on whether the function is above or below the x-axis, and the definite integral provides a precise numerical value for this area.

Graphing Functions

Graphing a function involves plotting its values on a coordinate system, which visually represents the relationship between the input (x-values) and output (f(x)-values). For the function ƒ(x) = 2 - |x|, the graph will show a V-shape, indicating how the function behaves over the interval [-2, 4]. Understanding the graph is crucial for identifying the bounded region and calculating the area accurately.

Area Between Curves

The area between a curve and the x-axis can be found by integrating the function over the specified interval. If the function dips below the x-axis, the area is considered negative, and adjustments may be needed to find the total area. In this case, the area of region R is determined by integrating the function ƒ(x) = 2 - |x| from -2 to 4, ensuring to account for any sections below the x-axis.