Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.7.56

Chapter 6, Problem 6.7.56

Force on the end of a tank Determine the force on a circular end of the tank in Figure 6.78 if the tank is full of gasoline. The density of gasoline is ρ = 737 kg/m³.

Verified step by step guidance

Identify the physical principle involved: The force on the circular end of the tank due to the gasoline is caused by the hydrostatic pressure exerted by the fluid at different depths.

Recall the formula for hydrostatic pressure at a depth \(h\) in a fluid: \(P = \rho g h\), where \(\rho\) is the fluid density, \(g\) is the acceleration due to gravity, and \(h\) is the depth below the fluid surface.

Determine the shape and dimensions of the circular end of the tank to express the pressure distribution over its surface. Since pressure varies with depth, the force is found by integrating the pressure over the area of the circular end.

Set up the integral for the total force \(F\) on the circular end: \(F = \int P \ dA = \int \rho g h \ dA\), where \(dA\) is an infinitesimal area element on the circular surface and \(h\) is the depth corresponding to that element.

Express \(h\) in terms of the vertical coordinate on the circular end, set the limits of integration according to the radius of the circle, and perform the integration to find the total force. Remember to multiply by \(g = 9.81 \ \text{m/s}^2\) and use \(\rho = 737 \ \text{kg/m}^3\).

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.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It increases linearly with depth and is calculated as P = ρgh, where ρ is the fluid density, g is gravitational acceleration, and h is the depth below the surface.

Force on a Surface in a Fluid

The force exerted by a fluid on a submerged surface is the product of the pressure at that surface and the area. For curved or vertical surfaces, the pressure varies with depth, so the total force is found by integrating pressure over the surface area.

Density and Its Role in Fluid Statics

Density (ρ) is the mass per unit volume of a substance and directly affects the hydrostatic pressure in a fluid. A higher density fluid exerts greater pressure at a given depth, influencing the force on submerged surfaces like the tank end.

Recommended video:

11:07

Pumping Liquids Example 2

Related Practice

Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x,y = 2x+2,x = 2, and x=6; about the y-axis

views

Textbook Question

A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where 0≤h≤8 0 ≤ ℎ ≤ 8 . Find the volume of water in the bowl as a function of h. (Check the special cases h=0 and h=8.)

163

views

Textbook Question

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

y=x,y=2x, and y=6 ; about the y-axis

views

Textbook Question

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.

y = (x−2)³ −2,x=0, and y=25; about the y-axis

views

Textbook Question

Find the area of the following regions, expressing your results in terms of the positive integer n≥2.

Let Aₙ be the area of the region bounded by f(x)=x^1/n and g(x)=x^n on the interval [0,1], where n is a positive integer. Evaluate lim n→∞ Aₙ and interpret the result. br

views

Textbook Question

Suppose a cut is made through a solid object perpendicular to the x-axis at a particular point x. Explain the meaning of A(x).

views