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categoryالرياضيات schoolبكالوريوس event_available2026-07-15

السؤال

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16. Find the area of the surface cut from the paraboloid x² + y²-z = 0 by the plane z = 2. 17. Integrate f(x, y, z) = xy + 1 over the surface S, where S is the part of the paraboloid z = x² + y² that lies inside the cylinder x² + y² = 4. 18. Find the outward flux of the vector field F = xi - yj across the closed surface S which is the first octant part of the pyramid bounded by the coordinate planes and the plane 3x + 4y + z = 12. 19. Use the surface integral in Stokes theorem to calculate the circulation of the field F = x²y³i+j+zk around the curve C which is the intersection of the cylinder x² + y² = 4 and the hemisphere x² + y² + z² = 16, z≥ 0. 20. Use the surface integral in Stokes theorem to calculate the circulation of the field F = yi + xzj + z²k around C which is the boundary of the triangle cut from the plane x + y + z = 1 by the first octant, counterclockwise when viewed from above. 21. Use the Divergence theorem to find the outward flux of F = In (x² + y²)i- tan¹j +Zx² +y2 k across the boundary of the region D which is the thick - walled cylinder 1≤x² + y² ≤2, -1≤z ≤2. 22. Use the Divergence theorem to find the outward flux of F = (y-x)i + (z-y)j + (y-x)k across the boundary of D which is the cube bounded by the planes x = ± 1,y=± 1, and z = ± 1.

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