Skip to main content
Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.6.9
Capitolo 6, Problema 6.6.9

Find the area of the surface generated when the given curve is revolved about the given axis.


y=8√x, for 9≤x≤20; about the x-axis

Guida verificata passo dopo passo
1
Step 1: Recall the formula for the surface area of a curve revolved about the x-axis. The formula is: S = 2π ∫ab y √(1 + (dy/dx)2) dx, where y is the function being revolved, and dy/dx is its derivative.
Step 2: Identify the given function y = 8√x and compute its derivative dy/dx. Using the power rule, dy/dx = 8 * (1/2) * x-1/2 = 4x-1/2.
Step 3: Substitute y = 8√x and dy/dx = 4x-1/2 into the surface area formula. This gives: S = 2π ∫920 (8√x) √(1 + (4x-1/2)2) dx.
Step 4: Simplify the expression inside the square root. Compute (4x-1/2)2 = 16x-1, so the integrand becomes: S = 2π ∫920 (8√x) √(1 + 16x-1) dx.
Step 5: Set up the integral for evaluation. The integral is now ready to be computed either analytically or numerically: S = 2π ∫920 (8√x) √(1 + 16/x) dx. Break this into manageable parts or use numerical methods to evaluate the integral.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
5m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Surface Area of Revolution

The surface area of revolution is calculated by rotating a curve around an axis. The formula involves integrating the length of the curve multiplied by the radius of rotation. For a curve y=f(x) revolved around the x-axis, the surface area S can be expressed as S = 2π ∫[a to b] f(x) √(1 + (f'(x))^2) dx, where f'(x) is the derivative of f(x).
Video consigliato:
09:07
Example 1: Minimizing Surface Area

Parametric Representation

In some cases, curves can be represented parametrically, which involves expressing the coordinates of points on the curve as functions of a parameter. For the curve y=8√x, it can be useful to express x in terms of a parameter t, where t=x, to facilitate integration. This representation helps in calculating derivatives and applying the surface area formula effectively.
Video consigliato:
08:02
Parameterizing Equations

Definite Integral

A definite integral calculates the accumulation of quantities, such as area, over a specific interval. In this context, the limits of integration (9 to 20) define the section of the curve being revolved. Understanding how to evaluate definite integrals is crucial for finding the total surface area generated by the revolution of the curve around the x-axis.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = 1−x²,x = 0, and y = 0, in the first quadrant; about the y-axis

67
views
Domanda del libro di testo

Why is integration used to find the work required to pump water out of a tank?

105
views
Domanda del libro di testo

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis

88
views
Domanda del libro di testo

52–54. Force on a window A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls. Find the force on the following windows. 


The window is circular, with a radius of 0.5 m, tangent to the bottom of the pool.

28
views
Domanda del libro di testo

Find the area of the region described in the following exercises.


The region in the first quadrant bounded by y=x^2/3 and y=4

71
views
Domanda del libro di testo

Suppose f and g have continuous derivatives on an interval [a, b]. Prove that if f(a)=g(a) and f(b)=g(b), then ∫a^b f′(x) dx = ∫a^b g′(x) dx.

62
views