2. Teorema Stokes Pertemuan 25 - 26
Matakuliah : Kalkulus II
Tahun : 2008 / 2009
Teorema Stokes Pertemuan

3. 2 Sketch the region of integration, determine the order of integration, and evaluate the integral.
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4. 3. Find the volume of the solid whose base is the region in the xy-plane that is bounded by the parabola y = 4 - x 2 and the line y = 3x, while the top of the solid is bounded by the plane z = x + 4.
Ans : 625 / 12
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5. Evaluate the improper integral
Ans : 1
Sketch the region bounded by the parabola x = y - y2 and the line y = -x. Then find the region's area as an iterated double integral. Ans . 4/3
4 Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane 4x + 2y + 6z = 12. Evaluate one of the integrals.
Ans.
Ans. 6
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6. 5 Find the volume of the wedge cut from the cylinder x 2 + y 2 = 1 by the planes z = - y and z = 0
Ans. 2/3
6 Find the volume of the region in the first octant bounded by the coordinate planes and the surface z = 4 - x 2 - y.
Ans ( 128 / 15 )
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7. STOKE'S THEOREM
Stoke's theorem states that, under conditions normally met in practice, the circulation of a vector field around the boundary of an oriented surface in space in the directions counterclockwise with respect to the surface's unit normal vector field n equals the integral of the normal component of the curl of the field over the surface.
The circulation of F = M i + N j + P k around the boundary of C of an oriented surface S in the direction counterclockwise with respect to the surface's unit normal vector n equals the integral of × F · n over S. • •
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8. and Stoke's theorem becomes
NOTE: If two different oriented surfaces S1 and S2 have the same boundary D, then their curl integrals are equal: • •
NOTE: If C is a curve in the xy-plane, oriented counterclockwise, and R is the region in the xy-plane bounded by C, then d□ - dx dy an • •
and Stoke's theorem becomes •
Notice that this is the circulation-curl form of Green's theorem.
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9. Since it is in the xy-plane, then n = k and ( × F) • n = 2.
EXAMPLE 1: Calculate the circulation of the field F = x2 i + 2x j + z2 k around the curve C: the ellipse 4x2 + y2 = 4 in the xyplane, counterclockwise when viewed from above.
SOLUTION:
Since it is in the xy-plane, then n = k and ( × F) • n = 2. •
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10. We are working with the ellipse 4x2 + y2 = 4 or x2 + y2/4 = 1, so I will use the transformation x = r cos 0 and y = 2r sin θ to transform this ellipse into a circle. I will also have to use the Jacobian to find the integrating factor for this integral.
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11. SOLUTION: Using the shortcut formula
EXAMPLE 2: Calculate the circulation of the field F = y i + xz j + x2 k around the curve C: the boundary of the triangle cut from the plane x + y + z = 1 by the first octant, counterclockwise when viewed from above.
SOLUTION: Using the shortcut formula
where M = y, N = xz, and P = x2, I will find × F.
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12. The triangle that we are looking at from above is in the plane x + y + z = 1, and the vector perpendicular to the plane is p = i + j + k. • • •
Let f = x + y + z - 1, and since the shadow is in the xy-plane, let p = k. •
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13. When z = 0, then x + y = 1 or y = 1 - x. When y = 0, then x = 1
When z = 0, then x + y = 1 or y = 1 - x. When y = 0, then x = 1. Finally, we have to get rid of the z in the integrand, so solve x + y + z = 1 for z. z = 1 - xy •
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14. SOLUTION: Before we start to solve this problem, we need a fact from
EXAMPLE 3: Use the surface integral in Stoke's theorem to calculate the flux of the curl of the field F = 2z i + 3x j + 5y k across the surface r (r, θ ) = (r cos θ ) i + (r sin θ ) j + (4 - r2) k, 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π in the direction of the outward unit normal n.
SOLUTION: Before we start to solve this problem, we need a fact from
integration of parametric surfaces, and here is the fact.
FACT:
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15. Now apply this to • • • •
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16. • •
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17. STOKE'S THEOREM FOR SURFACES WITH HOLES
DEFINITION: A region D is simply connected if every closed path in D can be contracted to a point in D without leaving D. (See figure 1)
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figure 1

18. THEOREM: If at every point of a simply connected open region D in space, then on any piecewise smooth closed path C in D,
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19. Soal soal Divergence Use the Divergence theorem to evaluate
Bila F = ( x2z , – y , xyz )
dan S dibatasi oleh kubus : 0 < x < a , 0 < y < a , 0 < z < a
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20. Stokes Theorem Verify Stokes Theorem where
F = ( z – y , x – z, x- y ) dan S : z = 4 – x2 – y2 , 0 < z
Use the surface integral in Stoke's theorem to calculate the flux of the curl of the field F = 2y i + (5 - 2x) j + (z2 - 2) k across the surface r (ϕ,θ) = (2sin ϕ cos θ ) i + (2sin ϕ sin θ ) j + (2cos ϕ ) k, 0 ≤ ϕ ≤ π /2, 0 ≤ θ ≤ 2π in the direction of the outward unit normal n.
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