Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(a) A (―2)

Verified step by step guidance

1. Understand the problem: We are tasked with evaluating A(-2), where A(x) = ∫₋₂ˣ ƒ(t) dt. This means we need to calculate the definite integral of the function ƒ(t) from -2 to -2.
2. Recall the property of definite integrals: If the upper and lower limits of the integral are the same, the integral evaluates to 0. Mathematically, ∫ₐₐ ƒ(t) dt = 0.
3. Apply this property to A(-2): Since the limits of integration are both -2, the integral evaluates to 0.
4. Conclude that A(-2) = 0 based on the property of definite integrals.
5. No further calculations are needed as the result is determined by the fundamental property of definite integrals.

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 integral symbol and limits of integration, providing a numerical value that corresponds to the total area, accounting for areas above and below the x-axis. In this context, A(x) and F(x) are defined as definite integrals of the function ƒ(t) over specified intervals.

Area Function

An area function, such as A(x) or F(x), is a function that gives the accumulated area under a curve from a specific starting point to a variable endpoint x. It is defined as the integral of a function from a constant lower limit to x, allowing for the evaluation of how the area changes as x varies. This concept is crucial for understanding how to compute areas based on the graph provided.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if F is an antiderivative of f on an interval, then the integral of f from a to b is equal to F(b) - F(a). This theorem is essential for evaluating area functions, as it allows us to compute the definite integral by finding the antiderivative of the function and applying the limits of integration.