1. Find the volume of the solid bounded by the paraboloids z = 6 – x² – y² and z = 2x² + 2y².Use polar coordinates.

Q: 23. Use Stokes' theorem to evaluate Se F · dr, where F(x, Y, z) = (xy, yz, zar) and C is the…

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Q: 2. Evaluate the triple integral //y o 2 = y, the xy-plane, and the parabolic cylinder y = 1- r2. y…

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Q: SSL. xy dV, where T is the first-octant solid bounded by the coordinate planes and the upper half of…

A: Here we have to find  ∫ ∫ ∫T xy dV , where T is the first-octant solid bounded by the coordinate…

Q: 1. The solid in the first octant bounded by the coordinate planes and the surface z = 1- y – x2.

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Q: 2. Let D = 2xyā, + 2(x² + z²)d, + 4yzd, C/m² and evaluate surface integrals to find total charge…

A: given D→=2xy ax→+2x2+z2 ay→+4yz az→ it is known that ρ=∇·D→ it is known that…

Q: 1. Let E be the solid under the paraboloid 2=r'+y. above the ry-plane, and inside the cylinder +y =…

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Q: Integrate f(x, y, z) = 20xz over the region in the first octant (x, y, z 2 0) above the parabolic…

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Q: . Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = y², xy-plane and…

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Q: 5. Evaluate the surface integral S_(x+ y) ds ; o is the part of the plane 2x + y +z = 3 that lies in…

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Q: Evaluate the surface integral. z + x2y x²y) ds S is the part of the cylinder y² + z2 = 9 that lies…

A: We have to find the surface integral from the given condition

Q: 2)use the polar coordinates to evaluate the double integral : A) || Vx2 + y² dA , R is the disk x² +…

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Q: #2. Use polar coordinates to find the volume of the solid bounded by the plane z = 0 and the…

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Q: Evaluate the surface integral. y ds S is the part of the paraboloid y = x² + z² that lies inside the…

A: The projection of S onto the xz-plane is the disk D=x,z: x2+z2≤9. We have to calculate the surface…

Q: Use Stokes' theorem to calculate JJ (A) nds by A = 3yi-xzi+yz²k where S is the surface of the…

A: The given vector field is: A=3yi-xzj +yz2k The surface S is the paraboloid 2z=x2+y2 bounded by z=2.…

Q: 12. DETAILS SCALCET8 15.3.019. Use polar coordinates to find the volume of the given solid. Under…

A: Given that, z=x2+y2 and above the disk x2+y2≤49Now,let x=rcosθ , y=rsinθ⇒x2+y2=r2r2≤49 ⇒0≤r≤7z=r2…

Q: Evaluate the surface integral. z2 dS, S is the part of the paraboloid  x = y2 + z2  given by  0 ≤…

A: given the surface integral  z2 dS  where S is the part of the paraboloid x= y2+z2   given by 0≤x≤1…

Q: Evaluate where G is the solid in the first octant bounded by paraboloid : = + y, the cylinder r+ y 9…

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Q: Determine the centroid, C(X,9.2). of the solid formed in the first octant bounded by z+y- 16 = 0 and…

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Q: 1. Find the volume of the solid bounded by the paraboloids z = 6- x² - y and z = 2x? + 2y2.Use polar…

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Q: 4. Find the volume of the solid bounded by the plane z = 0 and the paraboloid z=1-x² - y². (if using…

A: Solution :-

Q: 2. Use polar coordinates to find the volume of the solid under the paraboloid z = x² + y² and above…

A: Let's find volume of solid by using polar coordinates.

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: - 9 + 3 + 3y and Find the volume of the solid bounded by the paraboloids z == z = 8 – 3x - 3y

A: To find volume of the solid bounded by the paraboloidsz=-9+3x2+3y2 and z=8-3x2-3y2

Q: #3. Evaluate the surface integral xz ds, where S is the part of the plane 2x +2y +z = 4 that lies in…

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Q: Evaluate the surface integral. (z + x2y) ds S is the part of the cylinder y2 + z2 = 16 that lies…

A: Given ∫∫S(z+x2y)dS where S is the part of the cylinder y2+z2=16 that lies between the planes x=0 and…

Q: Consider D to be the region below the paraboloid z = 3+x² + y² , inside the cylinder x + y² =1 and…

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Q: 3. Use polar coordinates to find the surface area of the paraboloid z = x² +y² – 5 that lies below…

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Q: Evaluate the the triple integral ||| y dV; G is the solid enclosed z = y, the xy-plane, and the…

A: Evaluate triple integral:- ∫∫∫y dv ∫∫∫y dv=∫01∫01-x2∫0y ydz dy dx ∫∫∫y dv=∫01∫01-x2 yz0y dy dx ∫∫∫y…

Q: Consider the solid Q that is generated in the first octant by the surfaces S, : 2x + 3y + 3z = 6 S,:…

A: The region S2 is clearly a parabola y2 = 2-x and it is symmetrical in the x-axis.

Q: 3. Calculate the volume of the closed body formed with the surfaces; "y?+ (z-1)? = 1, x=2 and x+z=5"…

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Q: Use polar coordinates to find the volume of the solid under the paraboloid z=x2+y2 and above the…

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Q: 15. Evaluate SLL:(z+1)dV where E is the solid enclosed by the paraboloid y = x2 + z² and the plane y…

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Q: Use polar coordinates to find the volume of the solid under the paraboloid z = x2 + y2 + 1 and above…

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Q: Determine the element of surface area dS in the special case of the paraboloid of revolution z = x2…

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Q: Evaluate the surface integral. z2 ds, S is the part of the paraboloid x = y2 + z? given by 0 < x < 1

A: Evaluate the surface integral.

Q: Determine the centroid, C(i, ỹ,z), of the solid formed in the first octant bounded by z+y - 16 = 0…

A: The given curves are, z + y - 16 = 0 and 2x2 - 2(16 - y) = 0 Since the bounded region lies in first…

Q: 1) Evaluate the triple integral SS, f(x,y,z)dv : B) f(x,y,z) = 3y² – 2z , Q is the tetrahedron…

A: given f(x,y,z) = 3y2-2z, Q=3x+2y-z=6 The region bounded by the equation Q is a tetrahedron cutting…

Q: A: Determine the mass and center of mass of the solid that bounded by the paraboloid z = 9y? + 9x?…

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Q: Consider the solid region E bounded by the paraboloids z = x^2 + y^2 and z = 36 − 3x^2 − 3y^2

A: We are given that the solid is bounded by the paraboloids z=x2+y2 and z=36-3x2-3y2. For (a), We need…

Q: a) Find and sketch the curve of intersection between the paraboloids z = a? + y, = 8 – a² – žy². b)…

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Q: Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = the planes x = 0, x =…

A: Here I am using simple tripple integration to solve this question.

Q: Find the surface integral (x + y²)dS when S is the cylinder x² + y² = 4,0 < z < 5.

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Q: 4. Consider the solid formed by the boundaries outside the elliptic paraboloid = z and inside the…

A: Solution: Given that elliptic paraboloid: x2+y2=zand ellipsoid: x2+y2+z2-2z=0

Q: Use polar coordinates to determine the volume of the given solid Under the paraboloid: z = 18 – 2x?…

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Q: calculate ∫ ∫ ∫Ty dv, being the T region inside the paraboloid z= 1/(x2+y2)the sphere x2+y2+z2=5

A: We need to calculate the triple integral ∫ ∫ ∫Ty dv, where T being the region inside the paraboloid…

Q: Let S be the solid obtained by rotating the region shown in the figure below about the y-axis. y…

A: The function of the curve is y = 6x(x-1)2.

Q: Use Stokes’ Theorem to evaluate F(x, y, z) = 3zi + 4x j + 2yk; C is the boundary of the paraboloid…

A: Use Stokes theorem

Q: 10. Use polar coordinates to find the volume of the solid under the surface z = (x² + y²)² and above…

A: The polar coordinate conversion is as follows x = r cos(θ) y = r sin(θ) x2 + y2 = r2 Under the…

Q: 2. Evaluate the triple integral / || y dV; G is the solid enclosed by the plane z = y, the xy-plane,…

A: Solution is given below;