91Ó°ÊÓ

An antiderivative, also known as an indefinite integral, is a fundamental concept in calculus. It represents a function whose derivative is the given function. In simpler terms, it's the reverse of differentiation. To find an antiderivative, we look for a function whose rate of change (derivative) matches the function provided in the problem.

In our exercise, we need the antiderivative of the linear function \( f(x) = 130.6 - 9.6x \). We find it by computing the antiderivative of each term:

• The antiderivative of a constant like 130.6 is simply 130.6 times \( x \), resulting in \( 130.6x \).
• For the term \(-9.6x\), the antiderivative is found by increasing the power by one and dividing by the new power. Hence, the antiderivative is \(-4.8x^2\).

The overall antiderivative of our function is then \( 130.6x - 4.8x^2 + C \), where \( C \) represents the constant of integration which we don't need for definite integrals.
This expression is used to evaluate the definite integral and find the area under the curve over a specific interval.

A piecewise function is a function built from multiple sub-functions, each sub-function applying to a certain interval. It's like having a function that changes its formula based on the value of \( x \). These are especially useful when dealing with real-world situations that have different rules in different scenarios.

In the provided problem, the function \( f(x) \) is defined piecewise. It is given by:

• \( 130.6 - 9.6x \) for \( 5 \leq x \leq 12 \)
• \( 0 \) for values of \( x \) outside this range

However, since we are interested in finding the area under the curve on the interval \( 6 \leq x \leq 10 \), we only focus on the part of the function \( 130.6 - 9.6x \). This demonstrates how piecewise functions can define more complex behaviors and adapt to different condition sets.
Understanding piecewise functions helps in evaluating definite integrals over specific intervals by selecting the relevant piece of the function.

The area under a curve between two points on the x-axis is computed using a definite integral. This represents the accumulated quantity, like total distance traveled over time, from a starting point to an endpoint. When computing the area under the curve using calculus, we are essentially summing infinitely many infinitesimally small rectangles under the curve.

To determine this area for the function \( f(x) = 130.6 - 9.6x \) from \( x = 6 \) to \( x = 10 \), we evaluate the definite integral:\[ \int_6^{10} (130.6 - 9.6x) \, dx \]
Using the antiderivative \( 130.6x - 4.8x^2 \), we calculate \( [130.6x - 4.8x^2]_6^{10} \).

• Calculate \( (130.6(10) - 4.8(10)^2) \) to get 826.
• Calculate \( (130.6(6) - 4.8(6)^2) \) to get 610.8.
• Subtracting these,\( 826 - 610.8 \), gives us 215.2.

The calculated value, 215.20, represents the total area under the curve between the specified interval. This process is vital in fields such as physics and economics, where such integrals are used to compute quantities over time or space.