Chapter 8.3, Problem 4E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.4.

To determine

To express: The hydrostatic force as an integral function using the Riemann sum.

To evaluate: the integral as the function of hydrostatic force.

Explanation

Given:

The height of the vertical triangle plate is 5 ft.

The width of the vertical triangle plate is 10 ft.

The vertical triangle plate is submerged in the depth of the water is 7 ft.

Calculation:

Consider the weight density of water is Î´=62.5â€‰lb/ft3 .

Draw the vertical triangle plate x-axis as shown in Figure 1.

Refer to Figure 1.

Calculate the width of the ith strip as follows:

wixiâˆ’2=105wi=2(xiâˆ’2)

The depth of the ith strip as follows:

di=xi

Calculate the area of the ith rectangular strip using the Riemann sum:

Ai=wiÎ”x (1)

Substitute 2(xiâˆ’2) for wi in Equation (1).

Ai=2(xiâˆ’2)Î”x

Calculate the pressure of the ith strip using the Riemann sum:

Pi=Î´di (2)

Substitute xi for di in Equation (2).

Pi=Î´xi

The hydrostatic force as an integral using the Riemann sum:

F=limnâ†’âˆžâˆ‘i=1nPiAi (3)

Substitute Î´xi for P and 2(xiâˆ’2)Î”x for A in Equation (3)

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