Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full water. (The weight-density of water is 62.4 pounds per cubic foot.) lb Trapezoid

Q: Find the mass of the solid bounded below by the circular cone z = a + y° and above by the hemisphere…

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Q: Locate the mass center of the homogeneous solid body whose volume is determined by revolving the…

A: here region is rotated about z axis to create the solid  z=Moment about r -axis volume of solid

Q: use cylindrical coordinates to find the mass of the solid Q of density ρ.

A: Given Mass of solid Q is Now, we use cylindrical polar coordinates to evaluate the integral.

Q: Let W be the solid of revolution generated by revolving Q about the x– axis. Find the volume of W,…

A: The integration within specific limits of the variable of integration is known as definite…

Q: Use the disk method to verify that the volume of a sphere is 4/3 πr3 , where r is the radius ?

A: So sphere is basically a circle that is rotated through the x-axis. We will suppose to cut a disk…

Q: Select the formula for the Volume of the cone 2. T:r·h+ 2·T• r2 1 T• p2 . h 3 O2. T. r2 . h T:r·h² +…

A: The volume of a cone of radius 'r' and vertical height 'h' is given by the formula     Volume of…

Q: What solid is formed when this net is folded? B D

A: there is no introduction for the above question.

Q: Find the mass of the solid bounded below by the circular cone z = V2 + y° and above by the…

A: Circular is z = x2+y2  , hemisphere z = 62-x2-y2 density px,y,z = x2+y2+z2 Consider the volume of…

Q: Find the center of mass of the solid S bounded by the paraboloid z = 2x2 + 2y? and the plane z = 6.…

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Q: Determine the mass and coordinates of the center of gravity of the unit cube with the density…

A: Given density is, ρ(x,y,z)=x+2y+3z Let's find the mass of cube,…

Q: Find the moment or center of mass of the wire, as indicated. 10) Find the moment about the x-axis of…

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Q: Find the real number b such that the volume of solid B (r, +

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Q: Introduce an appropriate coordinate system and find the center of mass of the planar lamina.

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Q: Determine the coordinates of the mass center of the welded assembly of uniform slender rods made…

A: Given data, Let ρ = density per unit length

Q: Use cylindrical coordinates to find the volume of the solid that lies between the paraboloid z = x²…

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Q: Find the volume of the ellipsoid y? a2 <1, 62 by applying a change of variables which transforms the…

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Q: Use polar coordinates to find the volume of the given solid. Under the cone. := V² + y² And above…

A: Consider the provided information, z=x2+y2 and above the disk x2+y2≤9. Use the polar coordinate to…

Q: Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape.…

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Q: Determine the z-coordinate of the mass center of the homogeneous paraboloid of revolution shown. x…

A: Using the cylindrical coordinates

Q: Find the total mass of two turns of a spring with den r(t) = 6cos(t)i + 6sin(t)j + 6tk Р(х, у, 2)…

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Q: Calculate the fluid force on a side of the plate in Figure 11(B), submerged in a fluid of mass…

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Q: Find the total mass of two turns of a spring with density p in the shape of the circular helix…

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Q: Find the coordinates of the center of gravity of the homogeneous wire showen below. The portion on…

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Q: Find the center of mass of a lamina as pictured below, with inner radius 3 and outer radius 5, with…

A: Given problem:-

Q: Calculate the moments My and My and the center of mass of a lamina with the given density and shape.…

A: Let's find.

Q: Calculate the volume of the sphere x^2 + y^2 + z^2 = a^2, using both spherical and cylindrical…

A: Given the equation of the sphere is       x2+y2+z2 = a2 ,where a is the radius of the sphere.

Q: Find the center of mass of the solid S bounded by the paraboloid z = 3x? + 3y? and the plane z = 3.…

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Q: The volume of the solid generated by revolving the curve x = *, about the y-axis between y = 0 and y…

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Q: Determine the volume generated by revolving the plane figures about line a-a. a y 3m 2m 2m 2m 2m 2m…

A: ----------------------------------------------------------------------------- Equation of line…

Q: Calculate the moments My and My and the center of mass of a lamina with the given density (p = 3)…

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

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Q: Use polar coordinates to find the volume of the given solid. Inside the sphere x² + y² + z² = 16 and…

A: To find the volume of the solid inside the sphere x2+y2+z2=16 and outside the cylinder x2+y2=9

Q: EXERCISES Determine the x-, y-, and z-coordinates of the mass center of the body constructed of…

A: To find- Determine the x-, y-, and z-coordinates of the mass center of the body constructed of…

Q: Find the center of mass of the lamina that has the given shape and density. у 3 х, х + у 3D 12, у…

A: Let m be the mass of the lamina, then center of mass of lamina is given by x¯ , y¯ =Mym , Mxm R is…

Q: Q5: A- Find the mass of the region cut from the sphere++d by the cone z= density &?? +y² if the

A: Mass is given by M=∬∫V δ dv

Q: ゴー aab

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Q: Find the total mass of two turns of a spring with density p in the shape of the circular helix…

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Q: A Find the volume of the solid enclosed by the cylinder x? + y = 1, above the xy plane, and under…

A: Solution by using double integral as follows :

Q: Ged the volume and centroid generated by revolving the given plane area about the indicated axis. 1.…

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Q: Find the mass and the center of gravity of the cylindrical solid that has density 8(x,y,z) = 9 – z…

A: To find the mass and the center of gravity

Q: The 43-kg homogeneous smooth sphere rests on the 32° incline A and bears against the smooth vertical…

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Q: Q1:Find the volumes of the solids generated by revolving the regions W x', y = 0, x = 2 4 y =

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

Q: Find the z-coordinate of the center of mass

A: In this question, by using triple integral to find the z- coordinate of center of mass.

Q: Find the volume of the portion of the ball r2 +y? + z2 < 2 that lies above the cone z = (Ans: (/2–…

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Q: Compute the volume of W using cylindrical coordinates.

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Q: Find the total mass of two turns of a spring with density p in the shape of the circular helix…

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Q: Use cylindrical coordinates to find the volume of the solid. Solid inside both x? + y² + z? = 64 and…

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Q: Locate the coordinates of the centroid/center of gravity. b h

A: The definition of centroid is x¯=∫xdA∫dA and y¯=∫ydA∫dA. Here dA=dxdy ∫dA=∫0h∫0bdxdy=∫0hbdy=bh…

Q: Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape.…

A:

Q: Use the disk method to verify that the volume of a sphere is (4/3)π r3 , where r is the radius.

A: Volume of sphere of radius r is (4/3) πr3

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.