a) Evaluate (-ri-yj+2z k) -n dS, where o is the portion of the paraboloid z = 3r + 3y? between the planes z = 0 and z = 6 with n, the unit normal pointed in the positive z-axis direction.

Q: Use Cartesian coordinates to evaluate JIJ y² dv 2 D where D is the tetrahedron in the first octant…

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Q: HW Find the valume af theSelid beunded abave the Planc 2= x+2y+2,belw the X-y plane and Taterally…

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Q: z dv, where E is enclosed by the paraboloid z = x2 + y² and the plane z = 4. Evaluate

A: see 2nd step

Q: 39. Find the surface area of the portion of the hyperbolic paraboloid r(u, v) = (u+ v) i + (u – v)…

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Q: Consider the PDE y Ur + a Uyy + Uz – Uy = r² + y? In what region of the xy-plane is this PDE…

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Q: a) Evaluate // (-xi- yj+2z k) · n dS, where o is the portion of the paraboloid z = 3x2 + 3y² between…

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Q: Show that the area of that portion of the surface of the paraboloid x² + y² = 4z included between…

A: Here we need to find area between given plane and x*x + y*y = 4z

Q: Compute x dS where S is the portion of the plane z=1+x+2y that lies over the square [0, 1] × [0, 1].…

A: We have to find the surface integral ∫sxds.

Q: S is the part of the paraboloid z = x² +y? below the plane z = 4 oriented upward and F(x, y, z) :…

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Q: 18. The volume of the solid that lies between the paraboloid z = 2x² + 2y² and the plane z = 6

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Q: Compute 3xy dS, where S is the surface of the tetrahedron with sides z = 0, y = 0, x + z = 1, and x…

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Q: Let S, be the part of the paraboloid z = a² + y² that is inside the cylinder a? + y? = 1, S2 the…

A: “Since you have asked multiple questions, we will solve the first question for you i.e. (c). If you…

Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…

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Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: Find the surface area of the plane ( 15x- ? 5z+20y=D35) if it is in the first octant

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Q: Evaluate // e² dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder E x² + y? = 3,…

A: Since we are given with the multiple number of questions and answered according to the guideline.

Q: Let B be the ball with center at the origin and radius 1 SSS 4(z² + y² + z²)dV = B

A: Explanation of the answer is as follows

Q: What is the projection of r(t) = ti + t4j + etk onto the xz-plane?

A: Given: r(t)= ti+t4j+etk

Q: a) Find the points at which the twisted cubic F(t) = ti + t} +tk intersects the plane 4.x +2y+z = 24…

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Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, z) = (2x²y² + lIn 2) i – ry° j – tanh(x² + y³)…

A: Given function  F(x, y, z) = (2x2y2+lnz)i -xy3j -tanh(x2+y2)k we have to find ∬σF.nds by using guass…

Q: Find the surface area of the paraboloid z =1-x² - y? that is above the xy-plane.

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Q: a) Evaluate [-ri-yj+2: k) - n ds, (-a i- where a is the portion of the paraboloid z = 3r + 3y…

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Q: Use Cartesian coordinates to evaluate JIJ y² dV D where D is the tetrahedron in the first octant…

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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: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

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Q: V9/ Compute I ,cxyz), uhere s is the part f the plane +x that lies inside the half - cyllnder…

A: Given: The Integral is ∬SxyzdS. The surface is the part of the plane S  z=y+x that is inside the…

Q: Calculate the positive value of m so that the x+y+2z=m plane is tangent to the spherical surface…

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Q: Consider the area of the part of the plane z = ax + by+C that projects onto some region D in the…

A: If the area of the region D on xy-plane is k. That is, ∫∫DdA=k. If the area of the some region of…

Q: Use Cartesian coordinates to evaluate JJ v² av where D is the tetrahedron in the first octant…

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Q: Consider the surface given by x2 + y? + z? = a². Show that every normal line to this surface passes…

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Q: Let E be the region in the plane between the two ellipses 4x² +9y? = 1 and 4x2 + 9y2 = 4. Evaluate…

A: Given that two ellipse 4x2+9y2=1 and 4x2+9y2=4 Ans given E be the region in the plane between these…

Q: Prove that (x+) ) is the rolution on the dipporen tial equation y'+ y-e 9= C,e*+ C2x + Cg, y" - 4 y"…

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Q: 3) Find the surface area of the portion of the paraboloid z = x2 + y2 below the plane z = 1. %3D

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Q: (b) Evaluate SSI, zdV, where D is the solid tetrahedron bounded by the four planes x = 0, y = 0,z =…

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Q: Integrate g(x,y,z) = x + 5y + z over the portion of the plane 4x + 8y + z = 8 that lies in the first…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder æ? + y² = 5, and…

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Q: Evaluate ffp(4xy + 1) dA, where R is the region in the upper half-plane bounded by the circles a? +…

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Q: The surface area of the cap cut from the paraboloid y + z? = 3x by the plane x = 2 is Select one or…

A: Given the paraboloid  y2+z2=3x  we have to find the surface area of the cap cut from the paraboloid…

Q: Which of the following is the least distance from the origin to the plane 2x + 2y + z = 1? Lütfen…

A: We have to find

Q: Let B be the ball with center at the origin and radius 1 SIS 4(z² + y² + z*)dv = B

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Q: b) Use Gauss' theorem to evaluate SS F · ndS where F = (x² – y)i + y°z²j+ (xy – yz³)k and o is the…

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Q: . Let F(x, y, z) = (x³, y³, z³). Use Gauss's theorem to find ffsF dS if S is the sphere x? + y? + 22…

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Q: Evaluate (xr + y² + =³)? dV, where 2 is the solid hemisphere (half of the Ω sphere) enclosed by x +y…

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Q: The surface area of the part of the paraboloid z=x+y that lies under the plane z=2 is: Z=,

A: Surface area is given as A=∬D1+δzδx2+δzδy2dA

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x? + y?, the cylinder x? + y? 5, and the…

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Q: The distance from a point S(xo, Yo, Zo) to a plane M: ax + by + cz + d = 0 is given by In PS| d :…

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Q: a) Evaluate (-x i - yj+ 2z k) - n dS, where o is the portion of the paraboloid z = 3r2 + 3y? between…

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Q: a) Evaluate //(-ri-yj+2: k) n dS, where a is the portion of the paraboloid : = 3r - 3y between the…

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Q: yz ds where o is the part of the plane x + y+ z = 1 that lies in the first octant.

A: Given The given integral ∫∫σyzds over the region x+y+z=1 that lies in the first octant can evaluated…

Q: 1) Find the volume of the solid that lies between the paraboloid z = 2x² + 2y² and the plane Z = 8.

A: The main objective is to find the volume of solid that lies between the paraboloid z=2x2+2y2 and the…