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) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

Verified step by step guidance

1. Identify the piecewise function \( f(t) = \begin{cases} -2t + 8, & \text{if } t \leq 3 \\ 2, & \text{if } t > 3 \end{cases} \).
2. To find \( A(6) \), calculate the area under the curve from \( t = 0 \) to \( t = 6 \).
3. For \( 0 \leq t \leq 3 \), integrate \( -2t + 8 \) with respect to \( t \).
4. For \( 3 < t \leq 6 \), integrate \( 2 \) with respect to \( t \).
5. Add the results of the two integrals to find the total area \( A(6) \).

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 curve y = f(t) and the t-axis from t = 0 to t = x. This concept is fundamental in calculus as it connects geometric interpretations with integral calculus.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In the given problem, f(t) is defined differently for t ≤ 3 and t > 3, which requires careful consideration when calculating the area A(6). Understanding how to evaluate piecewise functions is crucial for accurately determining the area under the curve.

Definite Integrals

Definite integrals are used to compute the area under a curve between two points. To find A(6), one must evaluate the integral of f(t) from 0 to 6, taking into account the different expressions of f(t) over the relevant intervals. This concept is essential for solving problems involving area functions and understanding the accumulation of quantities.