Q) S = the part of the paraboloid z = 10-x^2-y^2 above z = 1 F = [x^3, xy, z] Find the flux of F across the paraboloid with the outwards orientation.

Q: . Find the center of mass of a thin plate covering the region bounded below by the parabola y = x…

A: The given curve is, y=x2 and the line y=x with density δx=12x. Sketch the bounded region as below:

Q: find the center of mass of a thin plate of constant density d covering the given region. The region…

A: Given: y=2x2-4x and y=2x-x2 Graph:

Q: calculate th integral SSo(x+y) dydx where D is th reigon that deter mined in thees Curves : Y = x, Y…

A: Solution is given below:

Q: The solid bounded by the surfaces S1: 2x + 3z = 6, S2: y = 2, S3: (y-2)² = 2x -2, S4: x = 0, S5: y =…

A: To find- The Solid bounded by the surfaces S1 : 2x + 3z = 6, S2 : y = 2, S3 : y - 22 = 2x - 2, S4 :…

Q: Compute the surface area of the solid of revolution formed by rotating the bounded region formed…

A:

Q: (10) Sketch the solid in the first octant bounded by x + z = 1 and y+ 2z = 2. If 8(x, y, z) = x…

A: The given region lies in the first octant, so x≥0, y≥0 and z≥0. To get the points on the planes,…

Q: Compute the surface area of the solid of revolution formed by rotating the bounded region formed…

A:

Q: Find the moment of inertia about the z-axis of a thin shell of constant density δ cut from the cone…

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Q: Find the center of mass of a thin plate covering the region bounded by the curve y2 = 4ax and the…

A:

Q: Find the center of mass of a thin plate covering the region between the x-axis and the curve y =…

A:

Q: 2. Calculate the centroid of the thin plate R which bounded by a = 1, y = 0 and, x² + y² = 4.

A:

Q: Find  the Volumeof the largest rectangular box lying above the xy-plane, with y less that or equal…

A: considerlength of rectangular box =x width of rectangular box  =yheight of rectangular box =z

Q: What is the solid of revolution obtained when you revolve the region bounded by the curves y=x−1,…

A: We have given curves y=x-1,  y=0 and x=3 And we can plot graph of these curves as We can  see here…

Q: Find the surface area cut from the cone 2x2 + 2y2 = 5z2, z > 0, by the cylinder x2 + y2 = 2y

A: To find the surface area cut from the cone 2x2 + 2y2 = 5z2, z > 0, by the cylinder x2 + y2 = 2y

Q: Find the center of mass of a thin plate covering the region bounded below by the parabola y = x2 and…

A: Given:Find the center of mass of a thin plate covering the region bounded below by the parabola y =…

Q: The surface area of the solid generated by revolving the straight line 2y = x+1 bounded by x-1, x 3…

A:

Q: 22 Find the Volume of the Solid of revolution bound al by the circuit x*+y=16, aud the lineSX=0 and…

A: Equation of the circle is given as x2 +y2 = 16 x2 +y2 = 42 Where radius of the circle r is 4 unit…

Q: 4. Illustrate and find the center of mass and moment of inertia with respect to x and y for the…

A: y = f(x) =4-x2   only in First quadrant…

Q: how by integration that the centroid of a uniform semi-circular lamina of radius a from ne centre is…

A:

Q: A solid is formed by revolving about the y-axis, the area bounded by the curve x^3=y, the y-axis,…

A: Solution :-

Q: . The solid bounded by z = 1 – x², z = 5 – 2x², y = 0, and y – z = 6. |

A:

Q: If the areas bounded by the curve y=x is revolved about y=0, what is the moment of inertia of the…

A: Given curve are y=x, x=2 and y=0. We have to find the the moment of inertia of the solid formed with…

Q: How do you locate the center of mass of a thin plate bounded by two curves y = ƒ(x) and y = g(x)…

A: Given:How do you locate the center of mass of a thin plate bounded by two curves y = ƒ(x) and y =…

Q: Find the center of mass of a thin plate covering the region between the x-axis and the curve y =…

A:

Q: Find the centroid (x¯,y¯)(x¯,y¯) of the region bounded by: y=7x^2+2x,   y=0,   x=0,   and   x=9?

A:

Q: Suppose that a semicircular wire has the equation y= V25- x and that its mass density is 8 = b- y.…

A:

Q: Consider a thin plate of constant density d lying in the regionbounded by the graphs of y = √x and x…

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Q: 2 y=x¯,y= 2x

A: The objective of the question is calculate area of the bounded region .

Q: Set up the surface integral for upward Flux of the upper hemisphere of radius = 5 (z20) such…

A:

Q: A plate occupying the region 0 < x < 12, 0 < y < 2 has density 8 = 8 gm/cm². Set up two integrals…

A:

Q: 2. Set up the integrals for the work required to pump the water in the drum shaped solid to the goal…

A: Work done= Force * distance                     =(force due to Weight)* distance

Q: Calculate the volume under the elliptic paraboloid z = 4x² + 6y2 and over the rectangle R = [-2, 2]…

A:

Q: 4) Write the integral to calculate the outward flux through F = (-yx2, xy,-z) across the sphere with…

A:

Q: A semi-circular lamina of redius R is bounded by the curve x2+2=4 and the x- axis. Find (a) the…

A: Given A semi-circular lamina of radius R is bounded by the curve x2+2=4 and the x-axis.

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: Calculate the volume under the elliptic paraboloid z = 4x2 + 6y2 and over the rectangle R = [-2, 2]…

A: The solution for the above question is shown below.

Q: Consider a thin plate of constant density d lying in the regionbounded by the graphs of y = √x and x…

A:

Q: Find a center of mass of a thin plate of density 8 = 5 bounded by the lines y = x and x=0 and the…

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Q: 2. Calculate the centroid of the thin plate R which bounded by r = 1, y = 0 and, 2² + y² = 4.

A:

Q: 2. The solid bounded by y = 0, x = y², x = 4, and z = Octant (ie. x, Y, z > 0). 2x in the First

A:

Q: Find the center of mass of a thin plate of constant density & covering the given Region enclosed by…

A: Curve 1: y=x-x2 Curve 2: y=-x We have to find the center of mass.

Q: Find the flux of F=(x, y, z) across the solid that is enclosed by the hemi = 0. Assume an outward or…

A: The Gauss Divergence Theorem states that the volume integral of the divergence of a vector field A…

Q: 4. Consider a soup can of height 2 and radius 3, whose base (bottom) lies in the r- y plane and is…

A: Given- Consider a soup can of height 2 and radius 3, whose base (bottom) lies in the x-y plane and…

Q: 2. Set up the integrals for the work required to pump the water in the drum shaped solid to the goal…

A: set up the integral for w9rk requireded to pump the water in the drum shaped solid to the goal line

Q: Find the volume of the solid bounded by the paraboloids z = – 5+ 2x2 + 2y² and z = 3 – 3x? – 3y? -

A: Here we use the cylindrical coordinate system to solve the problem.

Q: A solid is formed by revolving about the y-axis, the area bounded by the curve x^3=y, the y-axis,…

A: To find the centroid of the solid formed by revolving the below-bounded region about the y-axis.…

Q: Find the line integral of z2 dx + y dy + 4y dz, where C consists of two parts, C, and C,. C, is the…

A: Given line integral ∫Cz2dx+ydy+4ydz, where C consists of two parts C1 and C2. C1 is the intersection…

Q: (2) Let S be the surface of the sphere of radius 4 centered at the origin oriented away from the…

A: We can apply divergence theorem to find this

Q: Find the flux of F = 5zi+ 3y²j+ 2xk through of the total surface of the closed cone y = Vx² + z²,…

A: Flux of vector F through a surface S  can be given by  flux= ∫∫SF.ds We will use Divergence theorem…