R Let R be the region enclosed by the graphs of g(x) = -2+3 cos x ) and h(x) = 6 – 2(x – 1)², the y-axis, and the vertical line x = 2, as shown in the figure above. (a) Find the area of R. (b) Region R is the base of a solid. For the solid, at each x the cross section perpendicular to the x-axis has area A(x) 1 Find the volume of the solid. x + 3 (c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 6.

Q: A) Use Green's theorem to compute the area inside the = 1. 2² y² ellipse + 19² 14² Use the fact that…

A: (A). Given,                            x2192+y2142=1,      use the fact that the area can be written…

Q: 5) Find the area of the region bounded by the following function using the polar form f(8) = r =…

A:

Q: 5. Find the exact area of each described region. a. The finite region between the curves x = y(y-2)…

A: a) Given curve are          g(x) = y (y-2)    And f(x) = -(y-1)(y-3) Now , For point of intersection…

Q: cos( (a) Evaluate the integral Sp FdA where F(r, y) = and D is the triangle with vertices (0,0),…

A:

Q: O 5. An omega-triangle has vertices 0 = (0,0), = (1,0) and P = (0,h) where h > 0. (a) Prove that the…

A: Step 1:

Q: Rachel deposits $12,000 into a fixed asset account that earns 6% compounded quarterly. What is…

A: Formula:      where FV = Future Value          PV = The principal amount (the initial amount of…

Q: Question 6 Prior to running the Boston Marathon, a runner's personal best pace for a race is 10…

A: Given that Prior to running the Boston Marathon, a runner’s personal best pace for a race is 10…

Q: R is the region bounded by the functions f(x) = sin(x) + 1 and g(x) = (2x/π) + 1. Find the area of…

A: Given f(x)=sin(x)+1 and g(x)=2xπ+1   Given R is the region bounded by the functions f(x)=sin(x)+1…

Q: If C is a circle of radius 7 centered at the point ( -5, -1 ), then evaluate $ (34 (3y - esin(x)) dx…

A:

Q: 5. Find the exact area of each described region. The finite region between the curves x = y(y-2) and…

A:

Q: Question 20 Set up the integral that calculates the area of the region enclosed by the circle x2 + y…

A: The relation between polar coordinates and Cartesian coordinates is given by . Area bounded…

Q: Using a graph, it can be shown that the roots of the equation x V4 – x² = 2 – x are x x 0.81 and x =…

A: Given curves, y=x24-x2,y=2-x and roots of the equation x24-x2=2-x are x=0.81,x=2

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.