4. (a) Sketch the (finite) region R bounded by the functions y = x² and y = 4 and find the points of intersection. (b) Find the volume of revolution generated when the region R is rotated about the line y = 5.

Q: consider the plane region R bounded by the graphs of y = 3/4(x + 3)2, y = 3/4(x − 3)2, y = 3 and y =…

A:

Q: 21. Find the volume of the solid obtained by rotating the region bounded by the given curves about…

A:

Q: 8 SET UP THE INTEGRAL THAT COULD BE USED TO FIND THE VOLUME OF THE TETRAHEDRON ENCLOSED BY THE…

A: We have to find the volume of the tetrahedron bounded by the co-ordinate planes and the plane…

Q: 7. (a) By evaluation of the volume integral find the volume of the region bounded by upper…

A: (a) V=∫x∫yzdydx

Q: Determine the volume of the solid lying under the graph of z = x + 3y

A:

Q: Let a and h be positive real numbers. Let E be the surface in R³ given by 4a.x = y? + z2. (a) Find…

A: Given 4ax=y2+z2

Q: Calculate the Centroid of the region bounded by y= x³₁ X-2 and the x-axis. a) using vertical strip

A:

Q: =4-2² and b) Find the volume of the solid bounded by the surface y = the planes y = 0, z = 0 and +z…

A:

Q: Find the volume of the region under the surface z = corners (0, 0, 0), (7,0,0) and (7, 7,0). 2 1+x²…

A:

Q: Sketch the region D = {(x, y, z): x2 + y2 ≤ 4, 0 ≤ z ≤ 4}.

A:

Q: c) Find the volume of the solid bounded from the top by the surface f(x,y) = 2x +y² and from the…

A:

Q: = x² + y² Find the volume of the region bounded above by the paraboloid z = x and below by the…

A: The formula to calculate the volume of the region bounded above by the surface z=fx,y and below by…

Q: Find the surface area of the portion of the cylinder x² + z² =4 above the rectangle R = {(x,y) |…

A:

Q: 2. Sketch the region bounded by the curves r = 1-y², x = 2+y?, y = 1, and y = -1. Additionally,…

A:

Q: Find the volume of the region under the surface z = corners (0, 0, 0), (4, 0, 0) and (4, 4,0). 7…

A: The given problem is to find the volume of the region under the given surface z=7/(1+x2) and above…

Q: Find the volume of the region bounded by the sphere x+y+z² =9 and the planes x=0, y=0 and z = 0.

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Q: 3)Sketch the region enclosed by the given lines/curves and find the area of the region. a) y =…

A:

Q: Let W be the region bounded by z = 9-y.y = 3x², and the plane z = 0. (a) Calculate the volume of the…

A:

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.