Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.6.90
Chapter 8, Problem 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.

Verified step by step guidance
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
12m
Was this helpful?

Key Concepts

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

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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
06:11
Fundamental Theorem of Calculus Part 1
Related Practice
Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

18. ∫ dx / (225 − 16x²)

Textbook Question

5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.

6. T(4)

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

11. ∫ sin²(3x) dx

Textbook Question

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:

sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]

sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]

cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]

68. ∫ sin(5x)sin(7x) dx

Textbook Question

76. Apparent discrepancy

Three different computer algebra systems give the following results:

∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).

Explain how all three can be correct.

1
views
Textbook Question

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

29. ∫ e⁻ˣ sin(4x) dx