Compute 3xy dS, where S is the surface of the tetrahedron with sides z = 0, y = 0, x + z = 1, and x = y. (a) Compute the area of the portion of the cone x2 + y2 = z? with z 2 0 that is inside the sphere x2 + y2 + z2 = 2Rz, where R is a positive constant. (b) What is the area of that portion of the sphere that is inside the cone?

Q: 1) Find the volume of the prism whose base is the triangle in the xy-plane bounded by the x-axis and…

A: To find the volume of the prism described we just need to evaluate the given double integral…

Q: 28. F(x, y, z) = x³i+y°j+z°k, and S is the sphere of radius 3 and centered at the origin.

A: divF→=ddxx3+ddyy3+ddzz3        =3x2+3y2+3z2        =3x2+y2+z2  by divergence theorem,…

Q: The surface area of the part of z = 2 + V4 – x² – y² that lies inside the paraboloid z x² + y² is

A: This is a problem of vector calculus.

Q: 1. What is the differential for the total surface area of a frustum of a cone with l as the slant…

A: Given slant height l. Let r and R be the radii of the frustum. Let h be the height of the frustum.…

Q: (9) Evaluate // 2² + y² + z²)³/? dV where D is the solid region in the first octant between the two…

A: Volume between the two spheres using spherical coordinates

Q: 4. Evaluate the integral f. x'y°dy – x°y'dx, where C is the square determined by the lines x = 0, y…

A: As per Bartleby's answering policy, we can answer only one question and here we have answered the…

Q: The volume of the solid generated by the curve y = e, between z = 2 and z = 6 about the z- azis is

A: Given: The curve is y=e-x   To determine: The volume of the solid generated by the curve y=e-x…

Q: 2. The solid outside the cylinder x² + y² = 1 that is bounded above by the sphere x² + y? + x? = 8…

A: The solution is given below

Q: Find the extreme values of ƒ(x, y, z) = x( y + z) on the curve of intersection of the right circular…

A:

Q: Evaluate the triple integral. 16.  ∫∫∫T  xz dV , where T is the solid tetrahedron with vertices…

A: The solid tetrahedron T with vertices 0, 0, 0, (1, 0, 1), (0, 1 ,1) and 0,0, 1. The solid region is…

Q: ind the volume of the solid that lies under the hyperbolic paraboloid z = 3y2 - x2 + 4 and above the…

A: Using the double integral formula.

Q: 3. Evaluate , (x + z)dS where the surface S is the first-octant portion of the cylinder y² + z² = 9…

A: Step:- 1 Since, we know Surface integrals ∫∫Sf(x,y,z)dS=∫∫Df(x,θ) rx×rθ dxdθ…

Q: Consider the solid E determined by the inequalities x² + y² + z≤ 4, and let S denote the surface…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find the volume of the solid below

A: Given that the solid bounded below by the xy-plane, on the side by the sphere having radius 2, and…

Q: The surface area of the surface generated by revolving y=1-x? for 0<x<1 about the x- axis is None s=…

A: surface area S =∫012πy.ds where ds=1+(dy/dx)2

Q: 3. Find the volume of the solid that lies between the paraboloid z = x² + y² and the sphere x2 + y²…

A:

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: (2) Use the transformaiton u = xy, v = x² – y? to find JJ (æ* – y*)e*³ dA where R is the region in…

A: Given: The region bounded by the hyperbola xy=1 &amp;3, x2-y2=3 &amp;4 Where u=xy and v=x2-y2

Q: III Find the area of the portion of the cylindrical surface y = 4- a2 in the first octant that is…

A:

Q: 2. Show that the surface area of the solid of revolution obtained by revolving the curve y = 1/x, x…

A:

Q: Find the volume of a right circular cylinder of radius a = 4.00 with a paraboloid of revolution…

A:

Q: The surface area of the surface generated by revolving y=1-x² for 0<x<1 about the x- axis is s=…

A: If you like the solution then please give it a thumbs up.... The Answer is:              S =…

Q: Evaluate fff (3x + 2y)dV, where S is the solid bounded above by the plane z=4, below by plane z=0…

A: ∫∫∫(3x + 2y) dV=∫∫∫(3x + 2y) dxdydz = ∫z = 04∫y = -13∫x =-11(3x + 2y) dxdydz = ∫z = 04∫y = -134y…

Q: The base of a solid is the region in the xy-plane between the the lines y = x, y = 4x, x = 1 and x =…

A:

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: 1. Sketch carefully the solid enclosed by the three cylinders x + y* – 1,x² + 2² = 1, and y + z² =…

A:

Q: 5. Evaluate Sl. x2 dV where is the solid that lies within the cylinder x² + y² = 4 above the plane z…

A:

Q: 3. Find the volume under the paraboloid 3z = 2.12 + 2y2, above the triangle enclosed by the lines y…

A: First we have to draw graph and find limit according to which we have taken strip along y . Find…

Q: 1. Use the cylindrical coordinate to evaluate ff. x² + y²dV _where G is the solid bounded by…

A:

Q: How do I set up the triple integral of the function xy2 -3z, where the solid is bounded by the…

A:

Q: The shape of a redwood tree (without any branches) can be modeled by an upside down paraboloid, f(x,…

A:

Q: Given o V4-r2 4-r2-y2 -2-V4-12 x dzdydx 0. Choose the correct triple integrals in cylindrical…

A: Given

Q: The solid S is obtained by rotating the region enclosed by y = f(x) 2 0, a = 5+ 5e6 1- e6 5+ 5e16 1-…

A: The given question is solved below.

Q: 10. Find the area of the paraboloid x² + y + z² = 2 by the plane y = 0. (Hint: Implicit Surface)

A:

Q: 1 Find the surface area of the part of the hyperbolic paraboloid z = (x² - y²) that lies inside the…

A:

Q: - Evaluate the integral 4-z²-y² 1 dz dy dx Jo 4-z²-y² x² + y² + 2² using spherical coordinates.

A: To evaluate ∫-22∫04-x2∫-4-x2-y24-x2-y2 1x2+y2+z2 dzdydx

Q: Given the area bounded by the parabola x² – 4x + 12y- 20 = 0, the x- axis and the lines x = 1 and x…

A: Sol

Q: 27) If the solid of revolution formed by rotating R2 about the y-axis is a cylindrical shell, then…

A: You have posted multiple questions. As per the instruction, we will be answering only the first one…

Q: need to find the limits of integration in terms of ρ, φ, and θ.

A:

Q: (b) Find the area of the surface obtained by rotating the region bounded by the curves 7 35xy 5y12+…

A: To find the area of the surface obtained by rotating the region bounded by the below curves about…

Q: 1. Compute the average height of the cone z = a> 0. r² + y² above the disk x² + y² ≤ a² for

A:

Q: area of the portion of the cylindrical surface y = 4 − x2 in the first octant that is between the…

A: Given: The cylindrical surface is, y=4-x2 ........(1) And the xy-plane is, x2+2x+y-z =4

Q: 1. Evaluate the integral: SSS, x² + y² + z² dV where D lies above the cone z = spheres x? + y? + z²…

A: Here we have to evaluate the integral value, where D lies above the cone z= √(x2+y2) and between the…

Q: Find the volume of the solid that lies under the hyperbolic paraboloid z = 3y2 – x2 + 2 and above…

A: To find the volume of the solid that lies under the hyperbolic paraboloid  z=3y2-x2+2 and above the…

Q: The surface area of the surface generated by revolving y 1-x? for 0Sr<1 about the x-axis is S=…

A: Rule- If the curve y=f(x), a≤x≤b is rotated about the x−axis, then the surface area is given by

Q: 2. Calculate the volume of the solid that is bounded by the surface x²+y²-4y%3D0, inferiorly by the…

A:

Q: 5. Find the volume generated by the equation x2 +y² – 4x - 6y –- 12 = 0 if it is rotated about the…

A:

Q: ind the average height of the (single) cone z = sqrrt(x2 + y2) above the disk x2 + y2&lt;= a2 in the…

A: Given equation of cone is, Bounded by the disk,

Q: 1. Find the surface area of the paraboloid z = 4 – a² – y², for z > 1 using the formula 2 dz A(S) =…

A: The given paraboloid is: z=4-x2-y2 ,        for  z≥1

Q: 4. Find the area of the surface formed by revolving the graph of f(x) = x² on the interval [0,…

A: