2. Find the volume of the solid obtained by revolving the indicated region about the given line. 2.a. The region is bounded by the curves y = x and y = 4x - x and is rotated about the line x-4. 2.b. The region is bounded by the curves y = csc x, x =,x = and y = 0 and is rotated about the x-axis.

2. Find the volume of the solid obtained by revolving the indicated region about the given line. 2.a. The region is bounded by the curves y = x and y = 4x - x and is rotated about the line x-4. 2.b. The region is bounded by the curves y = csc x, x =,x = and y = 0 and is rotated about the x-axis.

Mathematics For Machine Technology

8th Edition

ISBN:9781337798310

Author:Peterson, John.

Publisher:Peterson, John.

Chapter65: Achievement Review—section Six

Section: Chapter Questions

Problem 44AR: Solve these prism and cylinder exercises. Where necessary, round the answers to 2 decimal places...

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Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Question

find the volume of the soild obtained by revolving the indicated region about the given line. a. the region is bounded by the curves y=x² and y=4x-x² and is rotated about the line x=4, b. the region is bounded by the curves y=csc , x=π/2 and y=0, and is rotated about x-axis.

General Instruction: Show your solutions and box your final answer. NO ERASURES ALLOWED.
1. Application of Definite Integral
1. Find the area of the region baunded by the given curves.
1. a. y = cosx ,y = sin 2x ,x = 0,x = !
2
1. b. y = x -x,y= 2x
2. Find the volume of the solid obtained by revolving the indicated region about the given line.
2.a. The region is bounded by the curves y = x' and y = 4x - x and is rotated about the line x=4.
2.b. The region is bounded by the curves y = csc x, x = ,x = and y = 0 and is rotated about
the x-axis.
3. A tank that has the shape of a hemisphere with a radius of 4 feet is full of water. If the opening
to the tank is 1 foot above the top of the tank, how much work in foot-pounds is required to
empty the tank?
4 ft
1 f
4. It takes a force of 15 pounds to stretch a spring 6 inches beyond its natural length. In foot-
pounds, how much work is required to stretch the spring 8 inches beyond its natural length?
5. A heavy rope that is 30 feet long and weighs 0.75 pounds per foot hangs over the edge of a
cliff. In foot-pounds, how much work is required to pull all the rope to the top of the cliff?

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