91Ó°ÊÓ

In calculus, a **definite integral** is a fundamental concept that helps us determine the total accumulation of quantities, such as areas under curves. When dealing with a function like \(f(x) = 4x - x^2\) over the interval \([0, 4]\), the definite integral is crucial for finding the total area under this curve.To compute the definite integral, we use the integral notation \( \int_{0}^{4} (4x - x^2) \, dx \). This notation signifies the process of accumulating the area under the curve from \(x = 0\) to \(x = 4\). The limits, \(0\) and \(4\), indicate the boundaries of this interval.For practical purposes:

• The lower limit \(0\) marks where we start calculating the accumulated area.
• The upper limit \(4\) marks where we finish the calculation.

Once you evaluate the function over this interval using the definite integral, you get the total area, which is represented by \( \int_{0}^{4} (4x - x^2) \, dx = 32 - \frac{64}{3} = \frac{32}{3} \).

An **antiderivative** is essentially the reverse of differentiation; it finds a function whose derivative is the function we started with. In the context of finding the average value of a function, antiderivatives play a crucial role.For the function \( f(x) = 4x - x^2 \), our task is to find an antiderivative, which we compute step-by-step:

• The antiderivative of \( 4x \) is \( 2x^2 \), because the derivative of \( 2x^2 \) is \( 4x \).
• The antiderivative of \( x^2 \) is \( \frac{1}{3}x^3 \), since the derivative of \( \frac{1}{3}x^3 \) is \( x^2 \).

Thus, the complete antiderivative is \( 2x^2 - \frac{1}{3}x^3 \). We use this antiderivative to evaluate the definite integral and find the total area under the curve, which figures into the average value computation.

**Interval calculation** is an essential step while determining the average value of a function. It involves defining the specific section of the x-axis over which the function is evaluated—in this case, from \(0\) to \(4\).For the function \(f(x) = 4x - x^2\), and interval \([0, 4]\), this calculation unfolds through the following steps:

• Subtract the lower limit from the upper limit to find the interval width: \(4 - 0 = 4\).
• Use the formula for the average value, \( \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of the interval.

The result \( \frac{1}{4 - 0} \times \frac{32}{3} = \frac{8}{3} \) represents the average height of the function \(f(x)\) over the interval \([0, 4]\). Understanding interval calculations helps us visualize and comprehend the behavior of the function across this specific span of x-values.