Determine the most efficient calculus method needed to solve each part; include a sketch for each part. (0.2) (8,2) R1 R2 R3 (0,0) x=4y (8,0) 4. Use the figure (above) to answer parts (a) through (c) below. Find the volume obtained when the specified region is rotated about the given line. (a) Rotate R, about the y-axis.

Q: find the region endlased R by the cunes V=12-2y +1 and Y= -2+2x ャ4.

A: Answer

Q: The surface defined by z = f(x, y) = 2x + y hovers above the region defined by the graphic below.

A: For the given surface  F(X,y)=2x+y The region bounded by curves is shown in figure  To find the…

Q: ). Explain how we get the Cylinder by equivalent relation.

A: Please check the next step for details

Q: Sketch the region enclosed by the given curves. 8 y = , y= y 3 -3 -2 -1 2 3 -1 Find the area of the…

A: To sketch the region enclosed by the below curves. y=14x2y=8x2+4

Q: Find the area of the shaded region. У 3 x = y - 2 y = 1 x = ey -3 -1 y = -1 -3 3. 2. 2. 2.

A: To find the area of the shaded region which is bounded by the below functions.

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.