Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.


a. Find A(2) .
ƒ(t) =6 <IMAGE>

Key Concepts

Area Under a Curve

The area under a curve represents the integral of a function over a specified interval. In this context, the area function A(x) calculates the total area between the t-axis and the graph of the function y = f(t) from t = 0 to t = x. This concept is fundamental in calculus as it connects geometric interpretations with integral calculus.

Definite Integral

A definite integral is a mathematical representation that computes the accumulation of quantities, such as area, over a specific interval. It is denoted as ∫[a, b] f(t) dt, where a and b are the limits of integration. In the problem, A(2) can be found by evaluating the definite integral of f(t) from 0 to 2, which gives the area under the curve from t = 0 to t = 2.

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. In this case, to find A(2), one must evaluate the integral of the function f(t) at the upper limit of 2. Understanding how to evaluate functions and integrals is crucial for solving problems related to area functions in calculus.