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 6.6.15
Capitolo 6, Problema 6.6.15

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


y=4x−1, for 1≤x≤4; about the y-axis (Hint: Integrate with respect to y.)

Guida verificata passo dopo passo
1
Step 1: Recall the formula for the surface area of a curve revolved about the y-axis. The formula is: A = 2π ∫ x √(1 + (dx/dy)²) dy, where x is expressed as a function of y.
Step 2: Rewrite the given equation y = 4x - 1 to express x in terms of y. Solve for x: x = (y + 1)/4.
Step 3: Compute dx/dy by differentiating x = (y + 1)/4 with respect to y. The derivative is: dx/dy = 1/4.
Step 4: Determine the limits of integration for y. Since the curve is defined for 1 ≤ x ≤ 4, substitute these x-values into the equation y = 4x - 1 to find the corresponding y-values. For x = 1, y = 3; for x = 4, y = 15. Thus, the limits of integration are 3 ≤ y ≤ 15.
Step 5: Substitute x = (y + 1)/4 and dx/dy = 1/4 into the surface area formula. The integral becomes: A = 2π ∫315 ((y + 1)/4) √(1 + (1/4)²) dy. Simplify the expression inside the integral and proceed 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:
3m

Concetti chiave

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

Surface Area of Revolution

The surface area of revolution refers to the area of a three-dimensional surface created when a two-dimensional curve is rotated around an axis. To find this area, we typically use integral calculus, applying the formula that involves the arc length of the curve and the radius of rotation. In this case, since the curve is revolved around the y-axis, we will need to express the curve in terms of y.
Video consigliato:
09:07
Example 1: Minimizing Surface Area

Changing Variables in Integration

Changing variables in integration is a technique used to simplify the process of evaluating integrals. In this problem, we need to integrate with respect to y, which requires us to express x as a function of y. This often involves solving the original equation for y and determining the limits of integration based on the given x-interval.
Video consigliato:
Percorso guidato
06:35
Changing Geometries

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specified limits. In this context, we will evaluate the integral of the surface area formula over the interval defined by the y-values corresponding to x = 1 and x = 4. Understanding how to set up and compute definite integrals is crucial for finding the total surface area generated by the revolution of the curve.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = y⁴/4 + 1/8y², for 1≤y≤2

97
views
Domanda del libro di testo

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.


y=2 sin x and y=0 on [0,π]; about y=−2

79
views
Domanda del libro di testo

Determine the area of the shaded region in the following figures.

184
views
Domanda del libro di testo

Force on a triangular plate A plate shaped like an isosceles triangle with a height of 1 m is placed on a vertical wall 1 m below the surface of a pool filled with water (see figure). Compute the force on the plate.

87
views
Domanda del libro di testo

Determine the area of the shaded region in the following figures.

127
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.

y = (1+x²)^−1,y = 0,x = 0, and x = 2; about the y-axis

128
views