A closed container is made from a right circular cylinder of radius r and height h with a hemispherical dome on top. Find the relationship between r and h that maximizes the volume for a given surface area s. (ans.:r=h =

Q: Find the volume of the 'ice-cream' that is bounded by x2 + y? + z2 99 and the upper cone z? = 3x2 +…

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Q: Let M be the region in the first quadrant bounded by the graph of y=3x+2, the graph of y=x+2, and…

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Q: Let B be the solid whose base is the circle x^2+y^2=49 and whose vertical cross sections…

A: Given, Circle- x²+y²=49=7² --------equation 1 The equation is of the form x²+y²=r². Therefore, the…

Q: Show by integration that the volume V of a right circular cylinder with base radius r and height h…

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Q: Find the volume of the 'ice-cream' that is bounded by x² + y2 + z² = 79 and the upper cone z? = 3x2…

A: We will use the triple integral to find out the volume, also we will use the cylindrical coordinate…

Q: If the area bounded by y = x² and y = 2 – x² is revolved about the x – axis and a vertical…

A: We have to find the volume of the solid if the area bounded by y=x2 and y=2-x2 is revolved about the…

Q: Given an area D bounded by curve y = 4x−x^2, line x+y = 6, and y-axis. Find the volume if the area D…

A: We are given an area D bounded by curve y = 4x - x2, line x + y = 6, and y-axis. We need to find the…

Q: A cylindrical can of height h and radius r is to be made with a fixed volume V. Show that the area…

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Q: Find the volume that is obtained by revolving the region about the r-axi 20 18+ 16+ 14+ 12+ 10+ y =…

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Q: Find the volume generated by the area bounded by the parabola x=√ay , and the line y=b that is…

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Q: Find the mass of the lamina described by the inequalities, given that its density is p(x, y) = xy. x…

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Q: find the mass of the lamina described by the inequalities, given that its density is ρ(x, y) = x +…

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Q: A rectangle R of length ℓ and width w is revolved about the line L . Find the volume of the…

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Q: 2. Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides…

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Q: 3x Assuming the box has a fixed volume. What is the value of y in terms of x that will result

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Q: The region bounded by the curves x^2+4y=8 and (x-4)^2=4y+8 is revolved about the line y=3. Determine…

A: The region bounded by curves:

Q: 3]. , Let V = ar²h, where V the volume of a right circular cone of radiusr and height h. Show that…

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Q: A box with square base and no top has volume 10 cubic units. If æ is the width of each side of the…

A: The given data is: Volume of box= 10 cubic units Width of the square base = x units The value of x…

Q: Compute the volume of the solid w w w w ww obtained by revolving the graph of the function f(x)=7x°…

A: Volume of solid obtained by revolving is given as Volume=∫ab πfx2dx

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: What is the volume of the region bounded by the lines f (x) = x²,x = 2, and y 49 rotated about the…

A: Given data:     The given equation is: f(x)=x2. The second equation is: x=2. The third equation is:…

Q: Find the total mass M of a 2-m rod of linear density ρ(x) = 1 + x(2 − x) kg/m, where x is the…

A: Consider the given given information. ρx=1+x2-x kg/m   The mass of the rod is given by the formula.…

Q: The area in the second quadrant of the circle x2 + y2= 2 is revolved about the line y + 1 = 0. What…

A: This problem belong to circle and line section from geometry and we solve this using the second…

Q: 1. Calculate the volume bounded by y = x², y = 0 and x = 2 rotated about the specified line, using…

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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: Given that R is the area bounded by the curve y = 9 – x^2 and y = 5. Calculate the volume of the…

A: Given that R is the area bounded by the curve y=9−x2 and y=5. We have to calculate the volume of the…

Q: Find the volume (in cubic units) of the region bounded by the curves (y-1)' = x – 1 ,and the lines y…

A: We have to find the volume of the region bounded by curvesy-12=x-1 and lines y=-1 and y=2when…

Q: Let m be the least possible surface area of a closed circular cylindrical can whose volume is equal…

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Q: ind the exact volume of the solid generated by revolving the region bounded by the given curves…

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Q: 2. Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides…

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Q: The base of a solid is the unit circle x^2+y^2=1 , and its cross sections perpendicular to the…

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Q: Find the volume of liquid needed to fill a sphere of radius R to height h = R.

A: answer is in next step. please give a like!!

Q: Find the surface area of the volume generated when the following curves revolve around the x-axis. y…

A: To solve the following problem of surface area of the volume generated.

Q: Given a lamina that occupies the region outside the circle r = 2 and inside the cardioid r = 2 +…

A: Given a lamina that occupies the region outside the circle r = 2 and inside the cardioid…

Q: If the region bounder by the two curves is revolved about x = 4, what is the volume of the resulting…

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Q: Find the mass of the lamina described by the inequalities, given that its density is p(x, y) = xy.…

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Q: The volume of a right circular cone of radius r and height h is V = Tr²h. Show that if the height…

A: Here is the solution of the given problem.

Q: For each problem, find the volume of the solid that results when the region enclosed by the curves…

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Q: 1. Show by integration that the volume V of a right circular cylinder with base radius r and height…

A: We’ll answer the first part of this question. since due to complexity. Please submit the question…

Q: (3) Find the mass of the lamina described by the inequalities, given that its density p(x, y) = xy.

A: Given,             ρ(x, y)=xy and  x≥0, 3≤y≤3+9-x2

Q: Compute the ”volume” of the 4-dimensional unit ball, which is the area enclosed by the equation x^2…

A: Given that, x2+y2+z2+w2=1. The volume for n dimensional unit ball is, V(n)=V(n-2)×2πn For n=4,…

Q: If the volume of the region bounded above by z = a²-x² - y2, below by the xy-plane, and lying…

A: Please see the attachment

Q: Determine the h/r ratio that minimizes the cylinder area for each given volume

A: The volume, V, of a cylinder is defined as V=πr2h, The surface area, A, of a cylinder is defined as…

Q: Given that the area R is limited by the curve y = x + 1, y = −x + 1 and the line y = α, 0 ≤ α ≤ 1.…

A: Given the bounded region: Rotation about x axis Volume of solid of revolution = 2π3

Q: If we have a half-circle of radius r centered at 0, prove thatthe volume of the sphere resulting…

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Q: Find the dimensions of the open cylindrical tank where the volume inside is maximized and that the…

A: Let r, h  be the radius of the cylindrical tank. Now, Let S represent the  Surface area  of the open…

Q: A thin metal rod is oriented along the x-axis over the interval TT T If the density of. the rod is…

A: We will use integration to solve this

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.