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A definite integral is a fundamental concept in calculus used to calculate the area under a curve. It's particularly useful for finding the average value of a function over a specific interval. For instance, to determine the average value of the function \[ f(x) = 3x^2 - 3 \] over the interval \([0, 1]\), we need to find the definite integral of this function on \([0, 1]\).In general, a definite integral is represented as:\[ \int_a^b f(x) \, dx \]where \( a \) and \( b \) are the lower and upper limits of the interval, respectively. This formula computes the net area between the x-axis and the curve \( f(x) \).To find the average value of the function, follow these steps:

• Set up the integral using the limits of the interval.
• Calculate the antiderivative.
• Evaluate the integral by substituting the upper and lower limits.
• Divide by the length of the interval (\( b-a \)).

Understanding how to compute a definite integral is essential in finding an average value over an interval.

The antiderivative, or indefinite integral, is a crucial part of solving problems involving definite integrals. It's essentially the reverse process of differentiation. To solve the problem of finding the average value of \( f(x) = 3x^2 - 3 \), you're required to first find its antiderivative.Here's how to find the antiderivative:

• Identify each term of the function separately. For \( 3x^2 \), the antiderivative is \( \frac{x^3}{3} \times 3 = x^3 \).
• The constant term \(-3\) becomes \(-3x\) after taking its antiderivative.

Putting it all together, the antiderivative of \( f(x) = 3x^2 - 3 \) is \( x^3 - 3x \).Once you find the antiderivative, you can apply it to calculate the evaluated limits for the definite integral. This procedure ensures the precise calculation of the average value of the function.

Interval notation is a mathematical shorthand to describe subsets of the real number line. You'll often use it in calculus to specify ranges over which functions are integrated. In our example, \([0, 1]\) is used to indicate the interval from \(0\) to \(1\) including both endpoints.Here's a quick guide to understand interval notation:

• Square brackets \([\, ]\) indicate that an endpoint is included in the interval (closed interval).
• Parentheses \((\, )\) denote that an endpoint is not included (open interval).

The use of closed intervals such as \([0, 1]\) in calculating definite integrals ensures that calculations account for the full span of the function's behavior from \(x = 0\) to \(x = 1\).This notation is not only succinct but also crucial for properly setting up integrals and understanding where functions reach their average value.

Graphing a function gives a visual representation of its behavior and helps to understand the solutions to calculus problems. For the function\( f(x) = 3x^2 - 3 \),you are tasked with sketching it over the interval\([0, 1]\).Here are steps to effectively graph this quadratic function:

• Identify the key features: it's a parabola opening upwards because the coefficient of \(x^2\) is positive.
• Find the vertex, which in this case, is the minimum point when the function is expressed in its standard form.
• Determine where the function intersects with the y-axis; for this function, it crosses at \(y = -3\).

On graph paper or using software, plot these points and draw the symmetric parabola accordingly.Graphing provides insight into the shape and position of the function relative to the x-axis, which ties back into solving calculus problems as it shows how areas under curves are interpreted in practical uses like finding average values.