Calculate the integral I = = • (y² + z²) dx + (z² + x²) dy + (x² + y²) dz where I is the Intersecting line of the sphere x2 + y2 + z2 = 2Rx and the cylinder x2 + y² = 2ax (0 < a < R, z > 0).

Q: If a parametric surface given by r₁(u, v) = f(u, v)i + g(u, v)j + h(u, v)k and −4 <u≤ 4,-2 ≤v < 2,…

A: The surface is  and The surface area of the parametric surface given by r1(u,v) =

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A: Given surface is 9x2+4y2=30 and plane is  9x+4y-6z=0 Bounded region in first octane ⇒x≥0, y≥0, z≥0…

Q: Sketch the cylinder given by x = 4- Z*2（z square）in three-dimensional space.

A: Given: The equation is x=4-z2. To sketch: The given equation in three-dimensional space.

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A:

Q: Calculate the surface area of the z=x²+y surface on the triangular region with vertices (0,0), (1,0)…

A: Let z=fx,y. Differentiate the function fx,y with respect to x and y, to obtain fxx,y and fyx,y.…

Q: 3. Find the area of the part of the surface x 2z + y° - 3, that lies above the triangle in the…

A: Given, x=2z+y2-3, i.e. z=12x-y2+3. As it is known that the surface area is given as,…

Q: /Sketch the Surface Z=V9-X+y- and verify that fxg = Exx %31

A: Second order Partial Differentiation

Q: Evaluate (image) over the sphere D = { (x,y,z) | x2 + y2+z2 ≤ R2 }

A: Given integral is ∭x2+y2+z2xyzdxdydz over the sphere D = { (x,y,z) | x2 + y2+z2 ≤ R2 } Change…

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A: We know that two planes are parallel if their normal vector are parallel. That is their, gradient…

Q: / (2² + y²) do, where S is the portion of the sphere r? + y? + z² = 9 in the first octant that is…

A:

Q: Explain why C lies on the surface of the cylinder x² + y² = 1.

A:

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A: Given: The plane is: 2x+3y+z=3. The elliptic cylinder is: x225+y29=1. To Find: The surface area of…

Q: Describe the surface whose equation is given by x² + y² + z² + 6y - 2z = 0.

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Q: Find the equation of the tangent plane to the sphere x² + y2 + z² = 1 at the point (1/ V3, 1/V3, 1/…

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Q: Evaluate the integral // Fn ds where F = (2r+32)i-(rz+y)3+(y² +2)& and S is the surface of the…

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Q: ) Find the points on the ellipsoid 6x² + y² + 3z² = 4 where the tangent plane is parallel to the…

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Q: Suppose you head toward the xy-plane from the surface x? + y? – 2² = –1 at the point (1, 1, v3) by…

A: given,     x2+y2-z2=-1 at the point 1,1,3

Q: Find a parametrization of the curve of intersection of the surfaces z = x² – y² and z = x² + xy + 1.

A: The given curves are z=x2-y2 and z=x2+xy+1 To find the intersecting curve : Let us equate both the…