Average distance on a parabola What is the average distance between the parabola y = 30𝓍 (20 ― 𝓍 ) and the 𝓍-axis on the interval [0, 20] ?

Verified step by step guidance

1. Step 1: Recognize that the average distance between the parabola and the x-axis is calculated using the formula for the average value of a function over an interval. The formula is: $\frac{1}{/}\int f_{\left(}dx$, where [a, b] is the interval and f(x) is the function.
2. Step 2: Substitute the given parabola equation y = 30𝓍(20 − 𝓍) into the formula. Here, f(x) = 30𝓍(20 − 𝓍), and the interval is [0, 20]. The formula becomes: $\frac{1}{/}\int {\mathrm{30}}_{\left(}dx$.
3. Step 3: Simplify the parabola equation. Expand y = 30𝓍(20 − 𝓍) to get y = 600𝓍 − 30𝓍². This is the function to integrate over the interval [0, 20].
4. Step 4: Set up the integral for the average value. The integral becomes: $\frac{1}{/}\int (600x-30x^2)dx$, where the limits of integration are 0 and 20.
5. Step 5: Compute the integral. Break it into two parts: $\int 600xdx$ and $\int 30x^{2}dx$. Evaluate each term separately, apply the limits of integration, and divide the result by 20 to find the average value.

Key Concepts

Average Value of a Function

The average value of a continuous function over an interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This concept is essential for determining the average distance between the parabola and the x-axis, as it involves integrating the function that represents the distance from the x-axis over the specified interval.

Definite Integral

A definite integral computes the accumulation of a quantity, represented as the area under a curve between two points on the x-axis. In this context, the definite integral of the function y = 30x(20 - x) from 0 to 20 will provide the total area under the parabola, which is necessary for finding the average distance to the x-axis.

Parabola Properties

A parabola is a symmetric curve defined by a quadratic function, which can open upwards or downwards. The given function y = 30x(20 - x) describes a downward-opening parabola with its vertex at the maximum point. Understanding the shape and properties of parabolas is crucial for visualizing the distance from the x-axis and interpreting the results of the integration.