Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.
(c) ∫₅⁷ ƒ(𝓍) d𝓍

Verified step by step guidance

1. Observe the graph of the function ƒ(x) over the interval [5, 7]. The graph consists of two linear segments: one from (5, 3) to (6, 0) and another from (6, 0) to (7, -3).
2. The definite integral ∫₅⁷ ƒ(x) dx represents the net area between the graph of ƒ(x) and the x-axis over the interval [5, 7]. Positive areas are above the x-axis, and negative areas are below the x-axis.
3. Break the integral into two parts based on the segments of the graph: ∫₅⁶ ƒ(x) dx for the segment from (5, 3) to (6, 0), and ∫₆⁷ ƒ(x) dx for the segment from (6, 0) to (7, -3).
4. For the segment from (5, 3) to (6, 0), calculate the area of the triangle formed. The base is 1 unit (from x = 5 to x = 6), and the height is 3 units. Use the formula for the area of a triangle: Area = (1/2) × base × height.
5. For the segment from (6, 0) to (7, -3), calculate the area of the triangle formed. The base is 1 unit (from x = 6 to x = 7), and the height is -3 units (negative because it is below the x-axis). Use the formula for the area of a triangle: Area = (1/2) × base × height. Add the two areas, keeping in mind their signs, to find the total value of the integral.

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 ∫₅⁷ ƒ(𝓍) d𝓍 calculates the area between the graph of the function f(x) and the x-axis from x = 5 to x = 7.

Piecewise Function

A piecewise function is defined by different expressions based on the input value. In the given graph, f(x) consists of segments that change at specific x-values, creating distinct linear sections. Understanding how to interpret these segments is crucial for accurately calculating the area under the curve for the definite integral.

Area Under the Curve

The area under the curve in a graph of a function can be interpreted as the integral of that function. For piecewise functions, this area can be computed by breaking it into simpler geometric shapes, such as rectangles and triangles, and summing their areas. This method simplifies the evaluation of the definite integral by leveraging geometric properties.