Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.4.33

Chapter 5, Problem 5.4.33

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

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Recommended video:

7:42

Properties of Parabolas

Related Practice

Textbook Question

Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.

The region bounded by y = 6 cos 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = π

Textbook Question

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.

v = [1 / (2t + 1)] (m/s), for 0 ≤ t ≤ 8 ; n = 4

Textbook Question

Average value of the derivative Suppose ƒ ' is a continuous function for all real numbers. Show that the average value of the derivative on an interval [a, b] is ƒ⁻' = (ƒ(b) ―ƒ(a))/ (b―a) . Interpret this result in terms of secant lines.

Textbook Question

General results Evaluate the following integrals in which the function ƒ is unspecified. Note that ƒ⁽ᵖ⁾ is the pth derivative of ƒ and ƒᵖ is the pth power of ƒ. Assume ƒ and its derivatives are continuous for all real numbers.

∫ (5 ƒ³ (𝓍) + 7ƒ² (𝓍) + ƒ (𝓍 )) ƒ'(𝓍) d𝓍

Textbook Question

v = 2t + 1(m/s), for 0 ≤ t ≤ 8 ; n = 2

views

Textbook Question

Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length ∆𝓍? List the grid points x₀ , x₁ , x₂ , x₃ and x₄. Which points are used for the left, right, and midpoint Riemann sums?

views