Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = 𝓍³ on [―1, 1]

Verified step by step guidance

1. Step 1: Recall the formula for the average value of a function ƒ(𝓍) on the interval [a, b]. The average value is given by: $\frac{1}{(b-a)}\int f_x\left(x\right)dx$. Here, a = -1 and b = 1.
2. Step 2: Substitute the given function ƒ(𝓍) = 𝓍³ into the formula. The integral becomes: $\frac{1}{2}\int \left(x\right)x^{3}dx$, where the factor $\frac{1}{2}$ comes from $\frac{1}{(}$.
3. Step 3: Compute the definite integral of 𝓍³ over the interval [―1, 1]. The integral of 𝓍³ is $\frac{1}{4}{x}^4$. Evaluate this expression at the bounds ―1 and 1.
4. Step 4: Subtract the value of the integral at the lower bound (―1) from the value at the upper bound (1). This gives: $\left(\frac{1}{4}{1}^4\right)-\left(\frac{1}{4}{-}^1\right)$.
5. Step 5: Multiply the result of the definite integral by $\frac{1}{2}$ to find the average value of the function. Finally, draw the graph of ƒ(𝓍) = 𝓍³ and indicate the average value as a horizontal line across the interval [―1, 1].

Key Concepts

Average Value of a Function

The average value of a continuous function over a closed interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This represents the mean value of the function across the specified interval, providing insight into the function's overall behavior rather than just its individual points.

Definite Integral

A definite integral computes the accumulation of a quantity, represented as the area under the curve of a function f(x) from a to b. It is denoted as ∫[a to b] f(x) dx and is fundamental in finding the average value, as it quantifies the total output of the function over the interval.

Graphing Functions

Graphing a function involves plotting its output values against input values on a coordinate system, which visually represents the function's behavior. For the function f(x) = x³, the graph will show a cubic curve, and marking the average value on this graph helps illustrate how the average compares to the function's actual values over the interval.