2) Determine the volume of the solid bounded by z = 4 – y, x + y = 0, x² = y, and the xy-plane.

Q: 8. Obtain the volume of the solid which is bounded by a circular paraboloid z= x +y´, cylinder x +y…

A: The solid is bounded by the curves, This represent that the solid is in first octant only.

Q: Find the volume of the solid that lies above the xy-plane and is bounded by the cylinder x² + y² = 1…

A: We are given that a solid lies above the xy plane and is bounded by cylinder x2+y2=1 and the plane…

Q: Find the volume of the solid whose base is the triangle in the xy-plane bounded by the x-axis, the…

A: Given plane z=3-x-y and the lines y=x and x=1. The objective is to find the volume of the solid…

Q: Let R be the region bounded by y=x, x= 3 and y = 0. The volume of the solid obtained by revolving R…

A:

Q: Find the volume of solid G bounded above by surface z= 8-x2 , bounded below by the surface z= x2 and…

A: Consider the given functions. z=8-x2,z=x2 Put the both surface equal. 8-x2=x22x2=8x2=4x=2 The volume…

Q: Find the volume of the solid bounded by: æy = 2, y = 0, æ = 2, and a = 4 a. about x = 1

A:

Q: Find the volume of the solid bounded by the xy-plane and the surfaces x2+y2=36 and z=x2+y2.

A:

Q: Find the volume of the solid bounded by the plane z = 0 and the paraboloid z = 25 - x2 - y2

A:

Q: Find the volume of the solid bounded above by the surface z = 1 – x² – y², below by the xy – plane,…

A:

Q: z = 0, y = 1, у %3D1, z = x² + y? y = x².

A:

Q: Compute the volume of the solid bounded by the given surface.   z = x + 1, z = y − 6, x = y2 − 2, x…

A:

Q: 5. Use the disk method to find the volume of the solid generated by revolving the region bounded by…

A:

Q: 2) Find the volume of the solid under the plane z = 3x +y and above the region determined by y = x'…

A:

Q: Find the volume of the solid bounded by the plane z= y+2 and the cylinder x'+y = 4 above the…

A: Given that cylinder x2+y2=4and z=y+2 The volume is the region is, V=∭dxdydz Since z=y+2 so z varies…

Q: 2) Use the disk method to find the volume of the solid generated by revolving the region bounded…

A: Here the curves are y=x13 and y=2.

Q: Find the volume of the solid bounded by the plane z =0 and the parabole z =1-x^2-y^2

A:

Q: Let R be the region bounded by x=y², x = 0 and y = 2 , then the volume of the solid formed by…

A: I have append the formula used in the solution part.

Q: Find the volume of the solid bounded by the planes x+y=1, x-y=1, x=0, z=0 and z=10

A:

Q: Compute the volume of the solid bounded by the surfaces x2+y2=33y, z=0 and z=Vx²+y2.

A: We will use of cylindrical coordinates to find the volume of solid

Q: find the volume of the solid bounded by the surface f(x,y) = 4 - 1/9 x^2 - 1/16 y^2, the planes x=2…

A: Integrals are generally used to calculate the area under the curve or volume of solids within a…

Q: Find the volume of the solid bounded z = x² + y², x² + y² = 1, and z = 0.

A: Use the cylindrical polar coordinates x=rcos θ.y=r sinθx2+y2=r2 Given, circular cylinder…

Q: Let R be the region bounded by y=x, x= 3 and y = 0. The volume of the solid obtained by revolving R…

A: Q.7)  Axis of revolution ( r(x) ) is line : x = 5  r(x)  =   (5 - x)

Q: Find the volume of the given solid. Bounded by the cylinders z = 2x², y = x² and the planes z = 0, y…

A: The given region is: z=2x2,  y=x2  and  the planes: z=0,  y=4

Q: Compute the volume of the solid bounded by the surfaces x²+y²=29y, z=0 and z=V x²+y².

A: We need to find volume.

Q: 44. Find the volume of the solid bounded above by the surface z = 1+xy and below by the triangular…

A: In mathematics, we study two types of figures, one is curvilinear while the other is…

Q: Find the volume of the solid bounded by paraboloid z =z+y and z *2-²+y and z = 4

A: To find the volume of the solid bounded by the below paraboloid and plane.

Q: Find the volume of the solid bounded above by the plane z = (1+x2+y2)2 and below by the plain z =0…

A: Let V be the volume of the solid which is bounded above by  z=21+x2+y22 and below by z=0 over the…

Q: Find the volume of the solid bounded by the cylinder x2 + y2 =16 and the planes y=z, x=0, z=0 in the…

A:

Q: Find the volume of the solid with the order dzdydx bounded by x2  + y2  = 4 and the planes x + z=2,…

A:

Q: 3. Find the volume of a solid G bounded above by surface z = 10², bounded below by the plane z = 1…

A:

Q: Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y +z = 7.

A:

Q: Find the volume of the solid bounded by the cylinder x² + y² = 4 and the planes y+z= 4 and z=0.

A:

Q: Calculate the volume of the solid bounded by x2+y2=4, z=0, and 4x+2y+z=16

A:

Q: Consider a solid bounded by, x² + y²=16, z=y², z=0 and x=0 Calculate its volume.

A: Introduction: Volume and the triple integral are equivalent. Integral is mostly used to calculate…

Q: 2) Find the volume of the solid under the plane z 3x + y and above the region determined by y = x'…

A:

Q: Compute the volume of the solid bounded E the given surface. z = x + 1, z = y - 15, x = y2 - 2, x =…

A: Triple integral is used to integrate the function of three variable over the three-dimensional…

Q: 8. Obtain the volume of the solid which is bounded by a circular paraboloid z=x +y´, cylinder x +y =…

A: Given equations are

Q: Find the volume of the given solid. Bounded by the cylinders z = 4x2, y = x2 and the planes z = 0, y…

A: Given data:   The first equation given is z=4x2. The second equation given is y=x2. The given value…

Q: y =, x = 1, x = 4, and y = 0 is revolved about the y - axis.

A:

Q: 4. Find the volume of the solid bounded by the planes given below y = 0, y = 1, x = 0, x = 1, z = 0…

A: We need to find the volume of the solid bounded by given planes. y=0, y=1, x=0, x=1, z=0 and x+y+z=3…

Q: Consider the solid bounded by the planes: z=x+y, z =12, x=0, y=0. Determine the volume of the solid.

A: Volume using the triple integral in rectangular coordinates is given by V=∬∫V dzdydx

Q: Calculate the volume of the solid bounded by the surfaces given by x2 + y2 = 1, z = 0 and z= x2 + y2

A:

Q: Find the volume of the solid bounded by the planes x 0, y z = 1 0, z = 0, and x + y

A: To calculate the volume of the three dimensional region given

Q: Compute the volume of the solid bounded by the surfaces x²+y²=29y, z=0 and z=Vx²+y2.

A: We need to find volume.

Q: 2 Find the volume of the solid bounded by the surface f(x,y) = 4 -x* -y?, the planes x = 2 and y =…

A:

Q: Calculate the volume of solid Q limited by surfaces z X2+ y2 = 1 and z = 0. + x2 + у2 1 Q 1+ x2 + y²

A: Consider the solid Q is bounded by the surfaces z=11+x2+y2,x2+y2=1,z=0. Find the volume of the solid…

Q: Find the volume of the solid generated when the plane region bounded by: y=4/x, x=1, x=4, and y=0…

A: Given regions are y=4/x, x=1, x=4 and y=0.

Q: Find the volume of the given solid. Bounded by the cylinders z = 3x2, y = x² and the planes z = 0, y…

A:

Q: Find the volume of the solid bounded by the xy-plane and the surfaces x2+y²=196 and z=x2+y².

A:

Q: Consider the solid bounded by the planes: =x+y, z=30, x=0, y=0. Determine the volume of the solid.

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.