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.




b. Find A(6).




ƒ(t) =6 <IMAGE>

Key Concepts

Area Under a Curve

The area under a curve in calculus represents the integral of a function over a specified interval. For a function y = f(t), the area A(x) from t = 0 to t = x is calculated using the definite integral ∫ from 0 to x of f(t) dt. This concept is fundamental in understanding how to compute the total area bounded by the curve and the axes.

Definite Integral

A definite integral is a mathematical tool used to calculate the accumulation of quantities, such as area, over a specific interval. It is denoted as ∫ from a to b of f(t) dt, where a and b are the limits of integration. The result of a definite integral is a number that represents the net area between the function and the t-axis over the interval [a, b].

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of the problem, evaluating A(6) means calculating the area under the curve from t = 0 to t = 6 for the given function f(t). This process is essential for finding specific values related to the area function A(x).