Bina Nusantara University 3 2 Sketch the region of integration, determine the order of integration, and evaluate the integral.
Bina Nusantara University 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
Bina Nusantara University 5 Evaluate the improper integral Ans : 1 Sketch the region bounded by the parabola x = y - y 2 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 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 )
Bina Nusantara University 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. STOKE'S THEOREM 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 8 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 9 EXAMPLE 1: Calculate the circulation of the field F = x 2 i + 2x j + z 2 k around the curve C: the ellipse 4x 2 + y 2 = 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 10 We are working with the ellipse 4x 2 + y 2 = 4 or x 2 + y 2 /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 11 EXAMPLE 2: Calculate the circulation of the field F = y i + xz j + x 2 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 = x 2, I will find × F.
Bina Nusantara University 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.
Bina Nusantara University 13 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 14 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 - r 2 ) 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 15 Now apply this to
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Bina Nusantara University 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) figure 1
Bina Nusantara University 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,
Bina Nusantara University 19 Soal soal Divergence Use the Divergence theorem to evaluate Bila F = ( x 2 z, – y, xyz ) dan S dibatasi oleh kubus : 0 < x < a, 0 < y < a, 0 < z < a
Bina Nusantara University 20 Stokes Theorem Verify Stokes Theorem where F = ( z – y, x – z, x- y ) dan S : z = 4 – x 2 – y 2, 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 + (z 2 - 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.