Determine by direct integration the volume of the paraboloid shown in the figure below [units^3] if b = 7.1.

Q: y2- 4y + 2x = 0, x = 2 and x-axis

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Q: Apply integration technique to find the volume generated by revolving the area bounded by the…

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Q: Find the volume of a pyramid with height 24 and rectangular base with dimensions 5 and 7 using…

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Q: Write the equation for the volume obtained by rotating the region shaded below about the line y =…

A: Hee, the volume formula is: V=π∫ab((f(x)-a)2-(g(x)-a)2) dx where the region bounded by the curves…

Q: By iterated integration, find the area bounded by the parabola y2 = 4ax and its latus rectum.

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Q: The volume generated by revolving the area bounded by the curve y = x² and y = 4 around the line y =…

A: We can use the formula π∫abR(x)2dx to evaluate the volume.

Q: write the Sum of the 2 Pterated integrals ntegral by converting to porar coordinutes Eraluate the…

A: Draw the regions from given limits of integrals and find the limits of added region as shown in…

Q: find the centroid of volume if the volum is equal to 7/15 π

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Q: calculate the volume of the 1st octant under the plan 2x + y + z = 10

A: answer

Q: Find the volume of the solid of revolution formed by rotating the area under y=9-x2 over [0, 3]…

A: Given curve is y=9-x2 ,[0,3]

Q: Determine the volume when the region, bounded by the given equations,

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Q: Find the volume of the solid cut from the first octant by the surfacez = 4 - x2 - y.

A: Consider the given surface  z=4-x2-y The domain is restricted by the first quadrant x≥0,  y≥0 The…

Q: Use a double integral to find the volume of the indicated solid.

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Q: the region bouded by x=y+2 and x=y^2 revolved about x=-1. Find the integral in finding the volume of…

A: The volume of the solid generated by revolving about the y−axis a region between the curves x = f(y)…

Q: (b) Interpret as a volume and calculate the integral without integrating.

A: As we know that Volume of a solid by integrtion is written as ∫∫∫dV=∬zdydx=∬xdydz=∬ydzdx

Q: Find the volume of a right circular cone with height 18 and base radius 9 using integration.

A: Take a cross-section area of thickness dx at a distance x from top, integrate it from 0 to 18.

Q: Find the volume generated if the area bounded by the curve y=x^2 and the line y=2x+3 is revolved…

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Q: Find the volume in the first octant between z=9-x^2-y^2 and the cylinder x^2+y^2=1.

A: The given surfaces can be plotted as follows: Thus, the required solid will be R=x, y, z: 0≤x≤1,…

Q: 4. Compute the volume rotating the area shown through one revolution about the x-axis. generated by

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Q: Find the volume of the solid whose base is bounded by the given function with the indicated cross…

A: 1. Given: x2+y2=4, quarter-circlesx=±2 are the limitsy=4-x2volume =π∫aby2dx

Q: Q8, Find the volume generated when the area bounded by:

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Q: Find the volume of a pyramid with height 29 and rectangular base with dimensions 6 and 9 using…

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Q: Find the volume of the solid formed by invoving the region by the graphs y=x^3 , y = 0 , x = 1 ,…

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Q: 3. Find the volume of a sphere with radius r by integration.

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Q: Compute the volume of the solid obtained by revolving the graph of the function f(x) = 7x5 on the…

A: Given function,

Q: A 6-inch-tall plastic cup is shaped like a surface obtained by rotating a line segment in the first…

A: Given: A 6-inch-tall plastic cup is shaped like a surface obtained by rotating a line segment in the…

Q: The area bounded by the curve y^2 = 12x and the line x = 3 is revolved about the line x =3. What is…

A: Recall: Formula for volume,    V=π∫ab(radius)2dy Given,  y2=12x Substitute x=3 in the above…

Q: The area bounded by the graphs of y = 2 - x^2 and y = x^2 revolves about the x-axis. Determine the…

A: The given curves are y=2-x2 and y=x2   Now find the point of intersection between the given curves.…

Q: Sketch the graph. Set up (DO Not Evaluate) an integral for the volume of the solid that results when…

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Q: The closed right circular cylindrical can of smallest surface area whose volume is 686r cm³ has a…

A: Topic = Mensuration

Q: For the pink region bounded by a parabola with vertex at (-1, 11 ) and the green line, solve for: a.…

A: The given equation of the parabola is y=-a(x-h)2+k, with vertex -1,11. ⇒y=-a(x+1)2+11 Given two…

Q: find the volume of the object obtained when one rotates the area under f(x) = cosx²)and over 0 <x< =…

A: Given, fx=cosx2  ,0≤x≤π2 The formula for finding the volume of the function fx is, ∫abπfx2dx

Q: What is the volume of the body of rotation formed by rotating the region bounded by the function y =…

A: Let us consider the given curve y=f(x) The volume of the solid of revolution about x-axis is…

Q: Use a double integral to find the volume of the solid bounded by the graphs of the equations

A: We have to find the volume  by using double integration

Q: Use cylinder shell method to graph and find volume: y=sqrt(x), y=0, x=1, about x=-1

A: If a region is bounded by a curve y=fx and x-axis on the interval a,b and it is rotating about a…

Q: What is the volume generated when the area bounded by y=2x2+1; x=1; x=2 is revolved about the line…

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Q: Write a short paragraph to discuss about the applications of integration to area?

A: Given that; To write a short paragraph to discuss about the applications of integration to area?

Q: Each integral represents the volume of a solid.Describe the solid.   ∫032 π x5 dx

A: Given : ∫03 2π x5 dx=?…

Q: Sketch the solid whose volume is given by the integral in the paper.

A: Given :                 The function is z = 1 - x - y            The region D is enclosed by the…

Q: Compute the volume of the solid obtained by revolving the graph of the function f(x)=8x$ on the…

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Q: Find the volume if the area bounded by the curve y = x³ + 1, the x-axis and the limits of x = 0 and…

A: (As per the guidelines, only 1st question will be answered).

Q: What is the volume of the rotator formed by the function y = 2-x ^ (2) and rotating the region…

A: Let us consider the curve y=f(x) The volume of the solid of revolution about x-axis is obtained by…

Q: Find the volume x=y-y2; x=0 revolving at y-axis

A: Given problem:- Find the volume x=y-y2 ; x=0 revolving at y-axis

Q: use a double integral to find the volume of the indicated solid.

A: Introduction: The volume beneath the surface z=f(x,y) over the region D can be deduced from the…

Q: Use a double integral to find the volume of the indicated nolid. 12 z- 12-X-y 11 10 4. 6. y 6 y-x

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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.