Suppose r ( t ) = 〈 t, 0 〉 , for a ≤ t ≤ b , is a parametric description of C ; note that C is the interval [ a , b ] on the x –axis. Show that ∫ C f ( x , y ) ds = ∫ a b f ( t , 0 ) d t , which is an ordinary, single – variable integral introduced in Chapter 5.
Suppose r ( t ) = 〈 t, 0 〉 , for a ≤ t ≤ b , is a parametric description of C ; note that C is the interval [ a , b ] on the x –axis. Show that ∫ C f ( x , y ) ds = ∫ a b f ( t , 0 ) d t , which is an ordinary, single – variable integral introduced in Chapter 5.
Solution Summary: The author explains that the integral displaystyle is an ordinary, single variable integral with a parametric description of C.
Suppose r(t) =
〈
t, 0
〉
, for a ≤ t ≤ b, is a parametric description of C; note that C is the interval [a, b] on the x–axis. Show that ∫Cf(x, y) ds =
∫
a
b
f
(
t
,
0
)
d
t
, which is an ordinary, single – variable integral introduced in Chapter 5.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (−4, 2), (−4, −3), (2, −2), (2, 7), and back to (-4, 2), in that order. Use Green's theorem to evaluate the
following integral.
Jo
(2xy) dx + (xy2) dy
Let C be the square with vertices (0, 0), (1, 0), (1, 1), and (0, 1) (oriented counter-clockwise).
Compute the line integral: y² dx + x² dy two ways. First, compute the integral directly
by parameterizing each side of the square. Then, compute the answer again using Green's
Theorem.
Assume that u and u are continuously differentiable functions. Using Green's theorem,
prove that
Uz
JE v|dA= [udv,
Uy
D
where D is some domain enclosed by a simple closed curve C with positive orientation.
Chapter 17 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
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