91Ó°ÊÓ

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. This process helps us determine the original function when its derivative is given. In our exercise, we deal with a piecewise function, where each section of the derivative, \( f'(x) \), needs a separate integration process.

For

• \(-1 < x < 0\), we are given \( f'(x) = 3x^2 \). Integrating this provides us with one part of the original function: \( F(x) = x^3 + C_1 \).


• For \(0 < x < 1\), \( f'(x) = 2e^x \) is given. Integration here leads us to another part of the function: \( G(x) = 2e^x + C_2 \).

Using integration allows us to form a complete picture of the function, piece by piece. It's crucial to handle each piece meticulously and provide continuity for a smooth transition at the boundary points.

Continuity in mathematics means that a function is unbroken and has no gaps over its specified domain. For a piecewise function like ours, achieving continuity requires that the pieces of the function connect smoothly without jumps or breaks.

A pivotal part of ensuring continuity is matching the values of the function at the points where the pieces meet. In our exercise:

• The left side of the function, \( f(x) = x^3 \), and the right side, \( f(x) = 2e^x - 2 \), both connect at \( x = 0 \).
• By verifying that both expressions equal zero at this point (\( f(0^-) = f(0^+) = 0 \)), we ensure the function is continuous at \( x = 0 \).

Thus, a piecewise definition of a function can lead to a continuous graph if the values at the junction points are correctly aligned, providing a seamless path through the entire domain.

An antiderivative is a function that reverses differentiation. It essentially helps us retrieve the original function from its derivative. When dealing with piecewise-defined functions, finding the antiderivative for each section is key to reconstructing the entire function.

For our exercise:

• The antiderivative of \( 3x^2 \) for the interval \(-1 < x < 0\) is \( x^3 + C_1 \).
• The antiderivative of \( 2e^x \) for \( 0 < x < 1 \) is \( 2e^x + C_2 \).

Each antiderivative has a constant added, \( C_1 \) or \( C_2 \), depending on the segment. These constants are determined using initial or boundary conditions, helping us perfectly fit each part into a coherent whole.

Boundary conditions are essential to solving problems involving differential equations or piecewise functions. These conditions allow us to tie different parts of the function together, ensuring a smooth and coherent definition across boundaries.

In this exercise, one such boundary condition was provided at \( x = 0 \), where \( f(0) = 0 \). This was critical for pinpointing the constants of integration and ensuring continuity:

• From the left, substituting into \( x^3 + C_1 = 0 \) at \( x = 0 \) yielded \( C_1 = 0 \).
• From the right, \( 2e^0 + C_2 = 0 \) at \( x = 0 \) provided \( C_2 = -2 \).

These constants ensure the seamless transition of the function across its domain, satisfying both continuity expectations and the given boundary condition. This is how boundary conditions unify the separate pieces of our function into a single, smooth element.