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**Teorema Stokes Pertemuan 25 - 26**

Matakuliah : Kalkulus II Tahun : 2008 / 2009 Teorema Stokes Pertemuan

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**2 Sketch the region of integration, determine the order of integration, and evaluate the integral.**

Bina Nusantara University

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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 Bina Nusantara University

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**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 Bina Nusantara University

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**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 ) Bina Nusantara University

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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. • • Bina Nusantara University

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**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. Bina Nusantara University

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**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. • Bina Nusantara University

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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. Bina Nusantara University

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**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. Bina Nusantara University

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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. • Bina Nusantara University

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**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 • Bina Nusantara University

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**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: Bina Nusantara University

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Now apply this to • • • • Bina Nusantara University

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• • Bina Nusantara University

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**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) Bina Nusantara University figure 1

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THEOREM: If at every point of a simply connected open region D in space, then on any piecewise smooth closed path C in D, Bina Nusantara University

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**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 Bina Nusantara University

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**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. Bina Nusantara University

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Section 16.3 Triple Integrals. A continuous function of 3 variables can be integrated over a solid region, W, in 3-space just as a function of two variables.

Section 16.3 Triple Integrals. A continuous function of 3 variables can be integrated over a solid region, W, in 3-space just as a function of two variables.

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