91Ó°ÊÓ

A piecewise function is a versatile tool in mathematics that allows us to define a function in segments, or "pieces," across different parts of its domain. This enables the modeling of more complicated scenarios where a single formula doesn't suffice. In the exercise, the function \( f(x) \) is defined separately on two intervals:

\[ f(x) = \begin{cases} x & \text{for } 0 \leq x \leq 1 \ x^2 & \text{for } 1 < x \leq 2 \end{cases} \]

This means:

• From 0 to 1, the function takes the form \( x \).
• From just over 1 to 2, it switches to the form \( x^2 \).

When solving problems involving piecewise functions, it's important to address each segment separately. This method requires careful integration over defined intervals to account for every part of the function's domain.

An antiderivative, also known as the indefinite integral, is essentially a function that "reverses" differentiation. It is the function \( F(x) \) such that \( F'(x) = f(x) \), essentially "undoing" the derivative. Finding an antiderivative is critical for using the Fundamental Theorem of Calculus, which links the concept of differentiation with integration.

In our problem, calculating the antiderivative of the piecewise function \( f(x) \) involves treating each piece separately:

• For \( 0 \leq x \leq 1 \), the antiderivative of \( f(x) = x \) is \( F(x) = \frac{x^2}{2} + C_1 \).
• For \( 1 < x \leq 2 \), the antiderivative of \( f(x) = x^2 \) is \( F(x) = \frac{x^3}{3} + C_2 \).

Where \( C_1 \) and \( C_2 \) are constants representing the family of antiderivatives. Ensuring continuity at \( x = 1 \) requires setting the left limit and right limit of the antiderivatives equal at this point, leading you to solve for \( C_1 \) and \( C_2 \).

The definite integral calculates the "accumulated" area under the curve of a function between two points, often describing the total accumulation of quantity, such as area under a curve between two x-values.
In the context of the Fundamental Theorem of Calculus, it connects the process of integration and differentiation.

For our piecewise function \( f(x) \) defined as:

• \( f(x) = x \) from 0 to 1.
• \( f(x) = x^2 \) from 1 to 2.

The definite integral of \( f(x) \) over the interval \([0, 2]\) involves performing separate integrations over each sub-interval:

1. Integrating \( f(x) = x \) from 0 to 1 gives \( \int_0^1 x \, dx = \frac{1}{2} \).

2. Integrating \( f(x) = x^2 \) from 1 to 2 yields \( \int_1^2 x^2 \, dx = \frac{7}{3} \).

Sum these to find the entire integral over \([0, 2]\): \( \int_0^2 f(x) \, dx = \frac{17}{6} \).

This result supports the verification of the Fundamental Theorem of Calculus by confirming that \( F(2) - F(0) \) equals this definite integral value.