Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. ∮ C ln 1 + y d x − x y 1 + y d y , where C is the triangle with vertices 0 , 0 , 2 , 0 , and 0 , 4 .
Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise. ∮ C ln 1 + y d x − x y 1 + y d y , where C is the triangle with vertices 0 , 0 , 2 , 0 , and 0 , 4 .
Use Green’s Theorem to evaluate the integral. In each exercise, assume that the curve C is oriented counterclockwise.
∮
C
ln
1
+
y
d
x
−
x
y
1
+
y
d
y
,
where C is the triangle with vertices
0
,
0
,
2
,
0
,
and
0
,
4
.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Evaluate This Integral
if curve C consists of curve C₁ which is a parabola y=x² from point (0,0) to point (2,4) and curve C₂ which is a vertical line segment from point (2,4) to point (2,6) if a and b are each constant.
Use this fact: The weight-density of water is 9800 newtons per cubic meter or 62.5 pounds per cubic foot.
Calculate (4x² + 5e³) dy, where C is the triangle of vertices (0,0), (2,0) and (2,2).
[(4x²+
Note: For a triangle, parameterize each side as a line segment, and calculate the line integral on
that side. Then, add up all of the answers for each side to find the final answer. Make sure to go in
order, from the first point to the second point, then from the second point to the third point, and then
from the third point back to the first point, because the order does matter.
5e) dy =
Chapter 15 Solutions
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