Ex. Find the volume V of the solid regron lyng a bove the xy-plane and beneath the poraboloid 2-1-x-y%.

Q: Set up the equation in solving volume of the solid generated when the arca bounded by z T. x = 0,…

A: This question is based on application of Integration.

Q: Find the volume of the solid that lies under the paraboloid == x² + y²₁ above the XY-plane, and…

A: Here integral transformed from cartesian to cylindrical co ordinate system . So 1st we need to know…

Q: 10. Show that the volume inside the ellipsoid ++=1 is 4abe (Hint: Use the change of variables x =au,…

A: We have to find volume of ellipsoid: Concept:

Q: 13. The volume generated when the region bounded by the curve x = /1+ y, the y-axis and the lines y…

A: Please rate

Q: Based on the figure below, find the volume of the solid generated by revolving area of the region…

A: Follow the procedure given below.

Q: (a) Consider the region bounded by the graphs of y=c(2x-4), x-1, x-3. and y 0 where c is just a…

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Q: Find the center of mass of a thin plate of density d = 3 bounded by the lines x = 0, y = x, and the…

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Q: 4. Find the volume of the solid generated by revolving the first quadrant area of the curves xy=2,…

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Q: (b) Find the volume under the paraboloid z-x+y above the triangle enclosed by the lines y=x,x-0 and…

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Q: (a) Sketch the region bounded by the graphs of y = x + 1, y = x2 – 1, x = 1, and x = 2. (b) Use the…

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Q: What is the volume, br^3, of the rotating body obtained by rotating the graph of the function y = Vx…

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Q: The region bounded by y = ca, a = 1 and the r-axis in quadrant I, is rotated 257 around the x-axis,…

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Q: The volume, when the curve y=v(3+ x^2 ) from x = 0 to x = 4 is rotated about x - axis is 100n 100/9…

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Q: O Find the volume of revolution (either by shells or Liashers) when the region enclosed bu by…

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Q: find the volume of the solid generated by revolv- ing the shaded region about the given axis. About…

A: Consider that a line x+2y=2 is rotated about x-axis in the interval x∈0,2. Solve the equation of…

Q: b) Find the volume of the solid which is in the first octant and bounded between the spheres p = 1…

A: To find- Find the volume of the solid which is in the first octant and bounded between the spheres ρ…

Q: Suppose that a thin metal wire in the shape of a semicircle occupies the curve c, where c is the top…

A: Given a thin metal wire in the shape of a semicircle occupies the curve C, where C is the top half…

Q: e Find the volume generated by revolving the area enclosed by the loop of the curve y² = x (x-3)²…

A: Solution is given below:

Q: 4. Find the volume created by the rotation round the x-axis of the part of the curve y² = 3x %3D…

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Q: The volume of the solid generated by the curve y = e¯ª, between x = 3 and x = 8 about the axis is -

A:

Q: 13. The region bounded by y = x°, y = 0, and x = 2 is revolved about the line x= 2. Find the volume…

A: Formula: The volume of the solid revolution generated by using disk method is given by,…

Q: 31 Find the volume of the solid generated by rotating the shaded region R through 360° about the…

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Q: 1 Find the volume generated when the area bounded by y=√√x and y=-x is rotated around the x-axis 2…

A: The region bounded by y=x and y=12x is rotated about the x-axis.

Q: Q2 (a) Find the volume under the surface z = 3x3 + 3x2y and over the rectangle R = {(x,y): 1 < x<…

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Q: B. Find the volume of the indicated solid i) under 2x - 3y + z = 6 and above the rectangle R = {(x,…

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Q: Find the volume of the solid under the paraboloid z=x2+y2 and above the disk x2+y2≤16.

A: Given: Paraboloid:  z = x2+y2 Disk:  x2+y2≤16 To find: Volume of the solid under the paraboloid z…

Q: 5. Find the volume under the paraboloid = -a? – y?, and above the cone z = - p² + y?.

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Q: Find the volume V of the solid below the paraboloid z = 6-x -y and above the following region.

A: We have to calculate ∬Rz dxdy

Q: Find the volume of the given solid. Under the paraboloid z = x² + 3y² and above the rectangle R= [0,…

A: Consider the provide question,

Q: (b) Find the volume of the solid generated when R is rotated about the x-axis.

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Q: The volume of the solid generated by the curve y = e, between r = 2 and a = 7 about %3D %3D the x…

A: Given : y = e-x

Q: D Evaluate the volume of the solid region E, in the first octant, that lies above z0 and below the…

A: Explanation of the answer is as follows

Q: d the volume of the solid whose base is the region enclosed by y = x² and y = 2, and the cross…

A: Topic:- application of integration

Q: Find the volume of the solid in revolution as it rotates about the y axis and the function 1. f(x) =…

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Q: 5. Find the volume of the solid of revolution formed by rotating the finite region bounded by curves…

A: Given :Curve :y=x-1 y=x-12

Q: Find the constant k>0 such that the volume of the region lying inside the sphere 2 x +y +z =a and…

A:

Q: 3. Find the volume of the solid that lies under the paraboloid 5z = x² + y², above the xy-plane, and…

A: To find the volume of the solid that is bounded by given curves.

Q: (b) Find the volume under the paraboloid z=x+y² above the triangle enclosed by the lines y=x, x=0…

A: To find the volume of the given solid.

Q: Q2:A) find the volume generated from rotation between curves : y = tan(x) y=1 and the interval (, )…

A: The volume of a solid obtained by rotating the region between the region y=f(x) and y=g(x) from x=a…

Q: (3+In x) Find the volume of the solid when the curve y=- is rotated 360° about the x-axis between x…

A: Using disk method volume is:  V=∫ee3πy2dx     =∫ee3π3+lnx3/2x1/22dx     =π∫ee33+lnx3xdx…

Q: 1. Find the volume under the parabolic f(x,y) = x²+ 4y² and above the rectangle R :0 < x < 1,1 < y <…

A: Given  fx,y=x2+4y2

Q: Use the cylindrical shell method to calculate the exact volume of the solid formed by rotating the…

A:

Q: 1) Calculate a)be the volume of the solid below the paraboloid 2+x²+(y-2)² and above the rectangle…

A: In this question, we find the volume of the solid bounded by paraboloid and base is a rectangle. we…

Q: a) Show that the volume enclosed when revolving the curve y = f(x) - where f: [a, b] → [0, 0) -…

A: Note- In the given question, it has to be V = π∫abf2x dx instead of V = π∫abf2x x. So, we are…

Q: O Find the volume of solid in the first octant bounded by the co- ordinate planes, the planes x-3,…

A: We will find out the required volume of the solid in the first octant.

Q: Calculate the volume of the solid of revolution generated by rotating the graph of the following…

A: It is given that f(x)=1x  ,x∈1,2 Volume of the solid of revolution generated by the rotating the…

Q: 3. Find the volume of the solid above the paraboloid z = x¯+y¯ –4 and under the R = {(r, 0) |1< r <…

A: We have to find the volume under conditions

Q: d. Compute the volume of the solid obtained by revolving the graph of on 10, al about the x-axis

A: In the given question,concept of volume of solids of revolution is obtained. Volume of Solids of…

Q: The volume, when the curve y = √3+ x2 from x = 0 to x = 4 is rotated about x-axis is O100m cu units…

A: Given:    y=3+x2 from x=0 to x=4  Volume of the curve V=π∫abfx2dx

Q: 5. Find the volume of the solid generated by revolving about the x-axis the smaller area bounded by…

A: Solution :- As per guidelines . I do first only.

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.