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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

When studying the formation of mountain ranges, geologists estimate the amount of work required to lift a mountain from sea level. Consider a mountain that is essentially in the shape of a right circular cone. Suppose that the weight density of the material in the vicinity of a point P is g(P) and the height is h(P).

(a) Find a definite integral that represents the total work done in forming the mountain.

(b) Assume that Mount Fuji in Japan is in the shape of a right circular cone with radius 62,000 ft, height 12,400 ft, and density a constant 200 lb/ft3. How much work was done in forming Mount Fuji if the land was initially at sea level?

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(a)

To determine

To find: The definite integral that represents the total work done in forming the mountains.

Explanation

Given:

The weight density of the mountain is g(P) .

The height of the mountain is h(P) .

Definition used:

“Total work done is the product of the mass of the given particle raised to the height and the weight density of that particle.”

Calculation:

By the definition stated above, the total work done is the product of the mass of the mountain raised to the height h(P) and the weight density of the mountain g(P) .

Mass of the mountain raised to the height is nothing but the volume of the mountain at the point P denoted by dV

(b)

To determine

To find: The work done in forming the Mount Fuji.

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Chapter 15 Solutions

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