Pertemuan Xxiii Xxiv

PERTEMUAN XXIII – XXIV

1. Sketch the region of integration and write an equivalent double integral with the order of integration reversed.

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

Ans : ( e8 – 1 ) /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 /

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
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 )

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.

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.
EXAMPLE 1: / Calculate the circulation of the field F = x 2i + 2x j + z2k around the curve C: the ellipse 4x 2 + y 2 = 4 in the xy-plane, counterclockwise when viewed from above.
SOLUTION:

Since it is in the xy-plane, then n = k and (F) n = 2.

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  and y = 2r sin  to transform this ellipse into a circle. I will also have to use the Jacobian to find the integrating factor for this integral.

EXAMPLE 2: / Calculate the circulation of the field F = y i + xz j + x 2k 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.
 = (0 - x) i + (0 - 2x) j + (z - 1) k = -x i -2x j + (z - 1) k
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.

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 - x- y
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,  ) = (r cos  ) i + (r sin  ) j + (4 - r 2) k, 0  r  2, 0  2 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: /
Now apply this to Fn d .

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
THEOREM: / If F = 0 at every point of a simply connected open region D in space, then on any piecewise smooth closed path C in D,

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

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 ( ,  ) = (2sin  cos  ) i + (2sin  sin  ) j + (2cos  ) k, 0  /2, 0  2 in the direction of the outward unit normaln.

TERIMA KASIH

XXIII- 1