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

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problema 6.7.46

Capitolo 6, Problema 6.7.46

46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.

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Identify the shape of the dam face, which is a rectangle with width 40 m and height 10 m, submerged in water up to the top.

Recall that the pressure exerted by a fluid at depth \(y\) is given by \(P(y) = \rho g y\), where \(\rho\) is the density of water, \(g\) is the acceleration due to gravity, and \(y\) is the depth measured from the surface.

Set up the coordinate system with \(y=0\) at the water surface and \(y\) increasing downward to 10 m at the bottom of the dam.

Express the force on a thin horizontal strip of the dam at depth \(y\) with thickness \(dy\) as \(dF = P(y) \times \text{area of strip} = \rho g y \times (40 \times dy)\).

Integrate this expression from \(y=0\) to \(y=10\) to find the total force: \(F = \int_0^{10} \rho g y \times 40 \, dy\).

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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 given by the formula P = ρgh, where ρ is the fluid density, g is acceleration due to gravity, and h is the depth below the surface.

Force on a Surface Due to Fluid Pressure

The total force exerted by a fluid on a submerged surface is found by integrating the pressure over the area. Since pressure varies with depth, the force is the integral of pressure times the differential area, often simplified by using the average pressure at the centroid of the surface.

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Pumping Liquids Example 2

Calculus Integration for Variable Pressure

Calculus integration is used to sum the infinitesimal forces over the dam face where pressure changes with depth. Setting up an integral with pressure as a function of depth and integrating over the height of the dam yields the total force.

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Fundamental Theorem of Calculus Part 1

Pratica correlata

Domanda del libro di testo

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.

v(t)=2 sin t, for 0≤t≤π

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Domanda del libro di testo

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

The region bounded by y=x^2−2x+1 and y=5x−9

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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=4−x^2,x=2, and y=4; about the y-axis

141

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Domanda del libro di testo

Look again at the region R in Figure 6.38 (p. 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.

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Domanda del libro di testo

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

The region bounded by y=√x, y=2x−15, and y=0

109

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Domanda del libro di testo

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

y = 2

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