Skip to main content
Ch. 8 - Integration Techniques
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. 8 - Integration TechniquesProblema 8.6.90
Capitolo 8, Problema 8.6.90

90. Work Let R be the region in the first quadrant bounded by the curve y = √(x⁴ - 4)
and the lines y = 0 and y = 2. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.

Guida verificata passo dopo passo
1
Step 1: Understand the problem. The region R is bounded by the curve y = √(x⁴ - 4), the line y = 0, and y = 2. This region is revolved about the y-axis to form a solid of revolution. The goal is to calculate the work required to pump all the water to the top of the tank, assuming the tank is full of water.
Step 2: Express x in terms of y. Since the curve is given as y = √(x⁴ - 4), solve for x in terms of y: x = (y² + 4)^(1/4). This will help us describe the radius of the solid at any height y.
Step 3: Set up the volume of a thin disk. A thin horizontal slice of the solid at height y has a radius x = (y² + 4)^(1/4) and thickness dy. The volume of this slice is dV = π * [x(y)]² * dy = π * (y² + 4)^(1/2) * dy.
Step 4: Calculate the work for a thin slice. The weight of the water in the slice is its volume multiplied by the density of water (ρ = 1000 kg/m³) and gravity (g = 9.8 m/s²). The work to pump this slice to the top of the tank (2 meters above the base) is dW = ρ * g * dV * (2 - y). Substitute dV from Step 3 into this expression.
Step 5: Integrate to find the total work. Integrate dW over the bounds of y from 0 to 2: W = ∫[0 to 2] ρ * g * π * (y² + 4)^(1/2) * (2 - y) dy. Simplify the integrand and compute the integral to find the total work required.

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:
12m

Concetti chiave

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

Solid of Revolution

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional area around an axis. In this problem, the region R is revolved around the y-axis, forming a tank. Understanding how to visualize and calculate the volume of such solids is crucial for determining the work needed to pump water from them.
Video consigliato:
04:48
Finding Volume Using Disks

Work Done Against Gravity

Work is defined as the force applied over a distance. In this context, the work required to pump water to the top of the tank involves calculating the gravitational force acting on the water and the distance each volume of water must be lifted. This requires integrating the product of the weight of the water and the height it needs to be raised.
Video consigliato:
Percorso guidato
05:40
Work Done On A Spring (Hooke's Law)

Integration in Calculus

Integration is a fundamental concept in calculus used to find areas, volumes, and other quantities that accumulate over a range. In this problem, integration will be used to sum the infinitesimal amounts of work done to lift each differential volume of water from the tank to the top. Understanding how to set up and evaluate these integrals is essential for solving the problem.
Video consigliato:
Percorso guidato
06:11
Fundamental Theorem of Calculus Part 1
Pratica correlata
Domanda del libro di testo

9–40. Integration by parts Evaluate the following integrals using integration by parts.

11. ∫ t · e⁶ᵗ dt

47
views
Domanda del libro di testo

9–40. Integration by parts Evaluate the following integrals using integration by parts.

26. ∫ t³ sin(t) dt

42
views
Domanda del libro di testo

7–64. Integration review Evaluate the following integrals.

57. ∫ dx / (x¹⸍² + x³⸍²)

57
views
Domanda del libro di testo

7–84. Evaluate the following integrals.

47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx

57
views
Domanda del libro di testo

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

54. ∫ y⁴/(1 + y²) dy

80
views
Domanda del libro di testo

9–40. Integration by parts Evaluate the following integrals using integration by parts.

36. ∫ from 0 to ln2 x eˣ dx

43
views