Question

Evaluate the Line Integral F = ?f only for a conservative vector field 14) F(x,y) = (1 + xy)e^{xy} i + x^2 e^{xy} j, C: r(t) = cos t i + 2 sin t j, 0 ? t ? ?/2 15) F(x,y,z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1,0,-2) to (4,6,3) 20) Show that the line integral ?_C sin y dx + (x cos y ? sin y) dy is independent of any path C from (2,0) to (1,?) and evaluate the integral.

          Evaluate the Line Integral

F = ?f only for a conservative vector field

14) F(x,y) = (1 + xy)e^{xy} i + x^2 e^{xy} j,   C: r(t) = cos t i + 2 sin t j,  0 ? t ? ?/2

15) F(x,y,z) = yz i + xz j + (xy + 2z) k,   C is the line segment from (1,0,-2) to (4,6,3)

20) Show that the line integral
?_C sin y dx + (x cos y ? sin y) dy
is independent of any path C from (2,0) to (1,?) and evaluate the integral.

Evaluate the Line Integral

F = ?f only for a conservative vector field

14) F(x,y) = (1 + xy)e^xy i + x^2 e^xy j, C: r(t) = cos t i + 2 sin t j, 0 ? t ? ?/2

15) F(x,y,z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1,0,-2) to (4,6,3)

20) Show that the line integral
?C sin y dx + (x cos y ? sin y) dy
is independent of any path C from (2,0) to (1,?) and evaluate the integral.

Added by Julie A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/10/2023

Step 1

First, we need to find the derivative of r(t) with respect to t: r'(t) = -sin(t)i + 2cos(t)j Now, we need to parameterize F in terms of t using r(t): F(r(t)) = (1 + (cos(t))(2sin(t)))e^{(cos(t))(2sin(t))}i + (cos(t))(2sin(t))e^{(cos(t))(2sin(t))}j Show more…

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