The base of a solid is bounded by y= x² and y= 4. All parallel cross – sections are squares perpendicular to the base and the y – axis. 5.

Q: The measurements x1 = 10m, x2 = 8m, and x3 = 6m %3D represent the sides of a rectangular…

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Q: Find the volume of the solid that lies under the elleptic paraboloid x2/9 + y2/16 + z + 1 and above…

A: Given the elleptic paraboloid x2/9 + y2/16 + z = 1  And rectangle R = [-1, 1] x [-3, 3] We have to…

Q: Find the surface area of the portion of the paraboloid z = 9- r - y? between the cylinders r?+ y? =…

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Q: Find the surface area of the portion S of the cone z^2=x^2+y^2 where z≥0z≥0, contained within the…

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Q: Consider the parallelepiped with adjacent edges u = 5i + 2j + k v=i+j+ 7k w = i + 9j + 5k Find the…

A: Given, u=5i+2j+k,v=i+j+7k,w=i+9j+5k.

Q: Find the surface area of ​​the solid intersection of the cylinders x² + z² = 1 and y² + z² = 1

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Q: Find the Volume cutoff from the cylinder x2 +y2 = 9 by the planes z = 0 and z = 5.

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Q: The surface of many airfoils can be described with an equation of the form xa₁x y ‚ = + 0 = ₂2 [²₁…

A: You can find matlab code and answer for this problem in next step

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: e surface generated by y=x^3 on [0,1], about the x-axis. Final Answer is A(s)=3.563 sq. units, Show…

A: Solution:- y=x3       0,1 y'=3x2 y'2=3x22y'2=9x4 Now find surface area S= ∫012πy1+y'2dx…

Q: Find the vo lume oF the solid OF revolution when the regiON bounded by y = x +1, the solid X-axis,…

A: Given problem:- Find the volume of the solid of revolution when the region bounded by y = x + 1 the…

Q: A solid is constructed between the graphs of y = f(x) and y = g(x) and the lines æ = a and æ = b so…

A: Given figure,

Q: Find the average height of the paraboloid z = x+y over the square 0sxs1, 0 sys1. The average height…

A: Here in this problem we are given                                      z=x2+y2  ,0≤x≤1 and 0≤y≤1We…

Q: 4 The area of the surface S = {(x, y, z) E R³ : z = 5 – z2 – y², z > 3} equals (A) 47 (B) 13 (C) (D)…

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Q: Evaluate //. dV, where E x2 + y² + lines between the spheres x + y + z? and x + y? + z2 = 25 in the…

A: Consider the provided question, Evaluate ∫∫∫E1x2+y2+z2dVwhere, E lines between the sphere x2+y2+z2=1…

Q: The solid bounded by x = y? – 1, z = 1 – y², z = 4 – x, and z = 0.

A: Given that,

Q: Find the area of the part of the cylinder y2 + z2 = 4 in the first octant, cut out by the planes x =…

A: Given. We are given a cylinder in the first octant.. y²+z² = 4 We need to find the volume of the…

Q: 4 The area of the surface S = {(x, Y, z) E R³ : z = 5 – x² – y², z > 3} equals (A) 4ñ 9. (B) 5* 13…

A: 4 Given S=x,y,z∈R3:z=5−x2−y2,z≥3.. We have to find the surface area of S At z=3 ⇒3=5−x2−y2⇒x2+y2=2…

Q: The volume of the body is given by the two surfaces sA z = 8 – 3(x² + y²), z = (x² + y²)² – 2 |

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Q: Compute the surface area of the domed structure D:= {(x, y, 2) ; x² + y² + (z – 1)² = 2 ; z > 0}.

A: x2 + y2 + (z-1)2 = 2  ,  x2 + y2 + (z-1)2 = 2  ,  z ≥ 0 This equation is for sphere of radius  = 2…

Q: Find the surface area of the portion of the surface x2- 2 In a+ V15 y - z = 0 above the rectangle 1…

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Q: (1) Find the area of the surface of the portion of the cone 2 +y² between the cylinder y? = x and…

A: To find the surface area of cone between given cylinder and plane, use the surface area formula as…

Q: Consider the solid that lies above the square (in the xy-plane) R = [0, 2] x [0, 2], and below the…

A: (a) For the lower left corner we have: (0,0)-(0,1)-(1,0)-(1,1) and evaluate the function at these…

Q: The volume of the solid given by the two surfaces is z = 8 - 3(x? + y*), z = (x? + y")2 - 2

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Q: nd the volume of the solid bounded by the paraboloid z = 32 – x² – y², the cylinder r² + y? = 16 and…

A: The solution are next step

Q: 9. Find the surface area of the part of the paraboloid a = 2 – 2? – y? that lies inside the surface…

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Q: The base of a solid is the region in the xy-plane between the the lines y = x, y = 4x, x = 1 and x =…

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Q: Find the volume of the solid that lies under the elliptic paraboloid x2/9 + y/16 + z = 1 and above…

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Q: 3. Find the volume of a solid enclosed by the parabolic cylinder y = x², the planes z = 0 and y +z =…

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Q: Suppose that the temperature in degrees Celsius at a point (x, y, z) of a solid E bounded by the…

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

A: Consider the provide question,

Q: The volume of the solid enclosed by the surface z = x/y and the rectangle 2sx5 4, 1syse² in the…

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Q: Find the volume of the solid that lies under the plane 4x + 10y – 2z + 19 = 0 and above the…

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Q: Find the surface area of the part of z = xy that lies in cylinder given by x2 + y2 =  1

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Q: 3) Find the volume of the solid enclosed by the paraboloid z = r + y?- 4 and the plane 2 = 5.

A: We have to find out the volume of given paraboloid equation   z2=x2+y2-4    with plane   z=5

Q: ind the area of the surface generated when the give curve is revolve about the x-axis    y=(x^3)…

A: Consider the given curve: Here the objective is to determine the area of the surface generated when…

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: area of the portion of the cylindrical surface y = 4 − x2 in the first octant that is between the…

A: Given: The cylindrical surface is, y=4-x2 ........(1) And the xy-plane is, x2+2x+y-z =4

Q: Find the area of the part of the surface ¤= (y/2 + z3/2) with y E [0, 2] and z E [0, 2]. Area =

A: For the solution of the problem follow the next step.

Q: The base of a solid is the triangular region bounded by the y-axis and the lines 3x + 2y = 6 and 3x…

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Q: 2. Calculate the volume of the solid that is bounded by the surface x²+y²-4y%3D0, inferiorly by the…

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Q: Consider the parallelepiped with adjacent edges u = 9i + 8j + k v =i+j+9k w =i+5j + 2k Find the…

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Q: Find the volume of the solid that lies under the plane 4x + 10y - 2z + 15 = 0 and above the…

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Q: Find the valume of the Solid bounded by the para boloid そ- 28-x2 152the cylinder x2+y?= 24 and xy-…

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Q: The volume of the tetrahedron bounded by the coordinate planes and the plane x+y+z=4 is 32 3 9. 2 O…

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Q: Find the volume of the wedge cut from the cylinder x2+y2=1 by the planes z = -y and z = 0.

A: The Volume of the wedge, where D is the solid and R is the region, is calculated as,

Q: (3) Consider a wire in the shape of a semi circle of radius 2, given by x² + y² = 4 and y ≥ 0.…

A: Formula: The mass of the region R with density function ρx, y is: m=∫∫Rρx, y dA The center of mass…

Q: Find the area of the part of the cone S = {(x, y, 2); z = Vx² + y?, x²+y² < 1}.

A: Given surface is S=x,y,z:z=x2+y2,x2+y2≤1 We have to find the area of the surface. Here, z=x2+y2…

Q: Q5: Find the volume of the solid bounded by surfaces z = /9 - x2 - y2 , and z = /25 - x2 - y? and…

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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.