تم الحل ✓
categoryالرياضيات
schoolبكالوريوس
event_available2026-07-15
السؤال
Transcribed Image Text:
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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