A. Find the volume of the region bounded above by paraboloid z = 5-x² - y and below by the paraboloid Z = 4X2 + 4y² B. Find the value of the constant B such that the vectors: u= Bi + 11j – 3k and v= 28i - ßj – 5k, are orthogonal

Q: 4. Find the surface area of the paraboloid S defined by the equation: (r, y, z) = (u cos v, usin v,…

A: We will find surface area by using formula.

Q: Consider AABC with its centroid at point G and median at BH. What is the length of GH if BG = 3x + 4…

A: Given Data: The length of BG is: BG=3x+4 The length of GH is: GH=x+10 The ∆ABC with its centroid at…

Q: Find the mass and center of mass of the S solid bounded by the paraboloid z = 8x? + 8y2 and the…

A: We have to find mass and enter of mass of the S solid.

Q: Let u = i−5k, v = 2i−4j + k,and w = 3i−2j + 5k. (a) u · (v × w) = ______. (b) The volume of the…

A: Given :           u = i - 5k , v = 2i - 4j + k, w = 3i - 2j + 5k

Q: 2. If Z is the centroid of ATUV, TZ = 60, XZ = 28, and WZ = 25, find each missing measure. a) ZV =…

A: Please see the explanation below

Q: If A = 2yz i - (x +3y - 2)j + (x²+z)k, evaluate |(V×A) •n dS over the surface of intersection of the…

A: Given:  A=2yzi-x+3y-2j+x2+zk, With the cylinder x2+y2=a2,x2+z2=a2 Now, calculating ∇×A.…

Q: Find the area of the surface. The part of the paraboloid r(u, v)= ucosvi+usinvj+u*k; 0sus8,0svS27 O…

A: Given r→u,v=ucosvi^+usinvj^+u2k^ and 0≤u≤8, 0≤v≤2π We have to find the area of the surface. Given…

Q: Let S be the solid tetrahedron bounded by the four planes a = 0, y = , z = 30, and z = 4x + y. !!…

A:

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: Find the area of the part of the hyperbolic araboloid z = y² – x² nat lies between the cylinders x²…

A:

Q: Evaluate F.N dS, where F(x, y, z) = 9x'i + 9xyj + 3x'y°k and S is the surface consisting of all…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4+ x² + y?, the cylinder E x2 + y? = 1, and…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y°, the cylinder z² + y² = 2, and…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5+x +y, the cylinder a2 + y = 1, and the xy…

A: The objective is to evaluate ∫∫∫EezdV, where E is enclosed by the paraboloid z=5+x2+y2, the cylinder…

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

A: We have to Evaluate the Integral using polar coordinate

Q: The outward flux of F(x, y, z) = xi + yj + zk across any unit cube is.

A: Given flux is F=xi+yj+zk. Compute div F. divF=∂∂xx+∂∂yy+∂∂zz=1+1+1=3

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: The volume of the solid bounded by the plane 3x+6y-2z=12 and the coordinate planes is O A. 2-2y+4 1…

A: We need to find the volume of the solid bounded by the plane 3x+6y-2z=12. Now,…

Q: Find the sum of maximum and minimum values of the function f(x,y, z) = 8x + 16y + 24z on the elipse…

A:

Q: Show that the value of ∮C xy2 dx + (x2y + 2x) dy around any square depends only on the area of the…

A:

Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

A:

Q: verify that (u * v) . w = (v * w) . u =(w * u) . v and find the volume of the parallelepiped (box)…

A: Concept:…

Q: If S is the solid bounded by the paraboloid z = 3x2 + 3y and the plane z = 4 (with constant…

A:

Q: Evaluate ez dV where E is enclosed by the paraboloid z = 4 + x? + y², the cylinder x? + y? = 1, and…

A:

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

A:

Q: ). Find the centroid of the first quadrant area bounded by the parabola y = x2 and the line y = x.…

A: According to the guidelines answer of first question can be given, for other please ask in a…

Q: Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u, v,…

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Consider the three points in R³ X1 = (-1,5,8), X2 = (1, 3, 7), X3 = (2,6, 4) and let a = x1 – X2, b…

A: Part (a) When two vectors is perpendicular then their dot product is 0.

Q: Find the area of the part of the paraboloid z=x² + y² that lies under the plane z=9.

A:

Q: In the (yz) plane consider the polygon A of vertices (3,0) (4, 2) (5, 0) and (4,-2) The volume of…

A: The volume of the solid of the revolution can be calculated using Pappu's theorem. The formula for…

Q: Find the volume of the parallelepiped spanned by the vectors i +j+ k, i –j+k and 3k.

A:

Q: Find the centroid of the region cut from the solid ball r2 + z2<= 1 by the half-planes u =…

A:

Q: Find the surface area of the portion of the part of the paraboloid z = 3 – x² – y² that lies above…

A: Given z = 3-x2-y2 , above the plane z=-1

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

A:

Q: Let C be the positively oriented square with vertices (0, 0), (2,0). (2, 2). (0, 2). Use Green's…

A: Green's Theorem:  For a positively oriented curve C∫CP dx+Q dy=∫∫D∂Q∂x-∂P∂y dAfor the region D…

Q: Find the average height of the paraboloid z = x2 + y2 over the square 0<= x <= 2, 0…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x² + y°, the cylinder 2² + y? 1, and the…

A:

Q: Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u = 2i…

A: We have to find the volume of parallelepiped using scalar triple product for the given adjacent…

Q: Let C be the positively oriented square with vertices (0,0), (3,0), (3,3), (0,3). Use Green's…

A:

Q: Evaluate € fff.₂²². e² dV where E is enclosed by the paraboloid z = 3 + z² + y², the cylinder z² +…

A: We need to find integral.

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

A:

Q: =valuate the double integral [2aydA, 2xy dA, where D is the triangular region with vertices (0,0),…

A: Given that a double integration and given  D is  triangular region with vertices (0,0) (1,2) and…

Q: In R, let S be the piece of the plane 3x + 3y + 3z = 3 that lies in the first octant, oriented…

A: Given S is the piece of the plane 3x+3y+3z=3 that lies in the first octant oriented upward. The…

Q: Q1- Determine the coordinates of the centroid of the area.: (1, 1) y= x,

A: The given problem is to find the centroid of the given shaded area of the region formed by given…

Q: SSx x dA; R is the portion of the disk x + ys 16 in the second quadrart

A: I use polar conversion method

Q: Find, by using polar coordinates in R2, the volume of the region bounded by the paraboloid z = 20 –…

A:

Q: Find the average haight of the Paraboloid z- xry* over the square osx <2, osd<2

A: We need to find the average height of paraboloid

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x + y°, the cylinder + y = 1, and the ry…

A: To Determine: evaluate ∫∫∫eZdV  where E is enclosed by the paraboloid z=5+x2+y2,the cylinder…

Q: For an area A in the x-y plane, the moment of inertia of the element dA about the x-axis is: O dlx =…

A: