Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.
(a) ∫₀⁴ ƒ(𝓍) d𝓍

Verified step by step guidance

1. Step 1: Observe the graph of the function ƒ(x) provided. The graph is piecewise linear, meaning it consists of straight-line segments. The integral ∫₀⁴ ƒ(𝓍) d𝓍 represents the area under the curve from x = 0 to x = 4.
2. Step 2: Break the interval [0, 4] into subintervals where the function is linear. From the graph, the function is constant at y = 2 from x = 0 to x = 2, and then increases linearly from y = 2 to y = 3 between x = 2 and x = 4.
3. Step 3: Calculate the area of the first region (x = 0 to x = 2). This is a rectangle with width 2 and height 2. The area of a rectangle is given by A = width × height.
4. Step 4: Calculate the area of the second region (x = 2 to x = 4). This is a trapezoid with bases of lengths 2 and 3, and a height of 2. The area of a trapezoid is given by A = (1/2) × (base₁ + base₂) × height.
5. Step 5: Add the areas of the two regions together to find the total area under the curve from x = 0 to x = 4. This sum represents the value of the definite integral ∫₀⁴ ƒ(𝓍) d𝓍.

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve between two points on the x-axis. It is calculated using the limits of integration, which define the interval over which the area is measured. In this context, the definite integral ∫₀⁴ ƒ(𝓍) d𝓍 calculates the area under the graph of the function f(x) from x = 0 to x = 4.

Area Under a Curve

The area under a curve can be interpreted geometrically as the total area between the curve and the x-axis over a specified interval. This area can be positive or negative depending on whether the curve is above or below the x-axis. In the given problem, the area can be computed by breaking it into geometric shapes such as rectangles and triangles, which can be easily calculated.

Piecewise Function

A piecewise function is defined by different expressions based on the input value. In the context of the graph provided, the function f(x) consists of linear segments that change at specific points. Understanding how to evaluate the function at different intervals is crucial for accurately calculating the definite integral, as each segment may contribute differently to the total area.