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] ?

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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_{(x)} d x

, where [a, b] is the interval and f(x) is the function.

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 30_{(x)} d x

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].

Step 4: Set up the integral for the average value. The integral becomes:

\frac{1}{/} \int (600 x - 30 x^{2}) d x

, where the limits of integration are 0 and 20.

Step 5: Compute the integral. Break it into two parts:

\int 600 x d x

and

\int 30 x^{2} d x

. Evaluate each term separately, apply the limits of integration, and divide the result by 20 to find the average value.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.