Question

Evaluate the line integral by the two following methods.

(xy dx + x^{2} dy)

*C* is counterclockwise around the rectangle with
vertices (0, 0), (2, 0), (2, 1), (0, 1)

(a) directly

(b) using Green's Theorem

Answer #1

Evaluate the line integral by the two following methods.
xy dx + x2y3 dy
C is counterclockwise around the triangle with vertices
(0, 0), (1, 0), and (1, 2)
(a) directly
(b) using Green's Theorem

Evaluate the line integral of
(x2+y2)dx + (5xy)dy over
the border of circle x2+y2=4 using Green's
Theorem.

Use Green's Theorem to evaluate the line integral.
C
(x2 −
y2) dx +
5y2dy
C: x2 +
y2 = 4

Evaluate the line integral:
(x^2 + y^2) dx + (5xy) dy on the edge of the circle: x^2 + y^2 = 4.
USING GREEN'S THEOREM.

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

Consider the sum of double integrals Z1 1/√2Zx √1−x2 xy dy dx
+Z√2 1 Zx 0 xy dy dx +Z2 √2Z√4−x2 0 xy dy dx .
a. [4] Combine into one integral and describe the domain of
integration in terms of polar coordinates. Give the range for the
radius r.
b. [4] Compute the integral.

Consider the sum of double integrals Z1 1/√2Zx √1−x2 xy dy dx
+Z√2 1 Zx 0
xy dy dx +Z2 √2Z√4−x2 0
xy dy dx .
a. [4] Combine into one integral and describe the domain of
integration in terms of polar coordinates. Give the range for the
radius r. b. [4] Compute the integral.

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