91Ó°ÊÓ

In calculus, piecewise functions are functions that have different expressions or rules for different parts of their domain. This allows a single function to behave differently depending on the input value, which helps in modeling real-world scenarios that are not uniform. In the given exercise, we see a piecewise function:

• For values of \( x \) that are less than or equal to 2, the function is defined as \( f(x) = 2 \).
• For values of \( x \) greater than 2, the function is \( f(x) = 3x \).

Each part of the piecewise function must be integrated separately within its specific interval. This means that for our exercise, we first compute the integral from 0 to 2 for \( f(x) = 2 \) and then from 2 to 4 for \( f(x) = 3x \). Breaking down the problem this way ensures that we accurately capture the behavior of the function across its entire domain.

The power rule is a fundamental technique in calculus used to find antiderivatives and integrals of polynomial functions. It states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) plus a constant of integration, usually denoted by \( C \). However, when dealing with definite integrals, the constant cancels out since we are calculating the net area between bounds.In the exercise, we apply the power rule in Step 2 while integrating \( 3x \) from 2 to 4. Since \( x \) is to the first power,\( n \) equals 1 in this case. Using the power rule, we perform the following steps:

• Find the antiderivative of \( 3x \). This gives us \( \frac{3x^2}{2} \).
• Evaluate this expression between the upper limit 4 and the lower limit 2, which calculates to \( \left[ \frac{3 \cdot 4^2}{2} \right] - \left[ \frac{3 \cdot 2^2}{2} \right] = 24 - 6 = 18 \).

This method efficiently provides the area under \( f(x) = 3x \) from \( x = 2 \) to \( x = 4 \).

Antiderivatives, also known as indefinite integrals, are functions whose derivatives recreate the original function. They are crucial for calculating definite integrals, which represent the net area under a curve between two points on the x-axis.When calculating a definite integral, such as in this exercise, the fundamental technique involves finding the antiderivative of the function and then performing the following:

• Evaluate the antiderivative at the upper limit of the integration interval.
• Subtract the value of the antiderivative evaluated at the lower limit.

For the piecewise function, two separate antiderivatives are calculated due to the change in the function's rule at \( x = 2 \):

• The integral of the constant \( 2 \), which is simply multiplied by the interval width to give \( 4 \).
• The antiderivative of \( 3x \), using the power rule, yields 18 as shown in the calculation.

Adding these results gives the complete area under the curve from \( x = 0 \) to \( x = 4 \), totaling \( 22 \). Understanding antiderivatives simplifies the process of computing definite integrals.