Question

Evaluate ∮C(x^3+xy)dx+(cos(y)+x2)dy∮C(x^3+xy)dx+(cos(y)+x^2)dy where C is the positively oriented boundary of the region bounded by C:0≤x^2+y^2≤16, x≥0,y≥0C:0≤x^2+y^2≤16,x≥0,y≥0

Answer #1

Use Green’s theorem to evaluate the integral: ∫(-x^2y)dx
+(xy^2)dy where C is the boundary of the region enclosed by y=
sqrt(9 − x^2) and the x-axis, traversed in the counterclockwise
direction.

Use Green's Theorem to evaluate the line integral along the
given positively oriented curve.
C
3y + 7e
x
dx +
8x + 3 cos(y2)
dy
C is the boundary of the region enclosed by the
parabolas
y = x2 and x = y2

Use Green's Theorem to evaluate the line integral along the
given positively oriented curve.
C
3y + 7e
x
dx +
8x + 9 cos(y2)
dy
C is the boundary of the region enclosed by the
parabolas
y = x2 and x = y2

Use Green's Theorem to evaluate the line integral along the
given positively oriented curve. C 3y + 5e x dx + 10x + 9 cos(y2)
dy C is the boundary of the region enclosed by the parabolas y = x2
and x = y2

solve diffeential equation.
( x2y +xy -y )dx + (x2 y -2 x2)
dy =0 answer x + ln x + x-1 + y- 2 lny = c
dy / dx + 2y = e-2x - x^2 y (0) =3 answer
y = 3 e -2s + e-2x ( intefral of e
-s^2ds ) s is power ^ 2 means s to power of
2

Evaluate double integral Z 2 0 Z 1 y/2 cos(x^2 )dx dy
(integral from 0 to 2)(integral from y/2 to 1) for cos(x^2) dx
dy

Evaluate C (y + 6 sin(x)) dx + (z2 + 2 cos(y)) dy + x3 dz where
C is the curve r(t) = sin(t), cos(t), sin(2t) , 0 ≤ t ≤ 2π. (Hint:
Observe that C lies on the surface z = 2xy.) C F · dr =

i)Please state if the following equations are exact or not:
(a) (sin(xy) − xy cos(xy))dx + x^2 cos(xy)dy = 0
(b) (x^3 + xy^2 )dx + (x^2 y + y^3 )dy = 0
ii) Determine if the following equation is exact, and if it is
exact, find its complete integral in the form g(x, y) = C:
(3(x)^2 + 2(y)^2 )dx + (4xy + 6(y)^2 )dy = 0

Use Green’s Theorem to evaluate the integral H C x ln ydx where
C is the positively oriented boundary curve of the region bounded
by x = 1, x = 2, y = e^x and y = e^(x^2) .

Evaluate the line integral of " (y^2)dx +
(x^2)dy " over the closed curve C which is the triangle
bounded by x = 0, x+y = 1, y = 0.

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