1.A 3-meter chain is hanging straight down the side of a building as shown at the bottom of the page. This chain has a variable density of p =x -3x+10 in kg/m. Acceleration due to gravity is 9.8m/s . we are interested in the work to pull all of the chain to the top of the building. (a) Label the sketch with the location of x 0. Your choice for the location of zero must be used for the remainder of the problem. (b) Write an expression for F(xr, ), the force acting on any small interval of chain. (c) Find the expression for the distance any representative part of the chain must travel (distance in terms of x, ). (The exact expression will depend on your location of zero in (a).) (d) Write the expression for W (x,) , the work to raise any small representative part of the chain. (e) Set up (but do not solve) the Reimann sum that approximates the total work done in lifting all of the chain. (f) Set up and solve the proper definite integral to find the total work done in lifting all of the chain to the top of the building. Include all steps of integration and include the proper work unit in your final answer.

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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.