Question

Line integral problem! Help is greatly appreciated! Problem is in the comments. Thanks!
[Calculus 3: Line Integrals]

Answer 1

Evaluate the line integral \[\int\limits_{C}^{ } (x^2 + y^2) dx + 2xy dy\]where C is the path of the semicircular arc of the circle \[x^2 + y^2 = 81\]starting at (9,0) and ending at (-9,0) going counterclockwise.

Answer 2

\(x^2 + y^2 = 81 \)

\(x \, dx + y \, dy = 0\)

\( y \, dy = -x \, dx\)


\(\implies \int\limits_{C}^{ } 81 dx + 2x \color{red}{(-x \; dx )} = \int_{ C} \; dx \quad (81 - 2x^2) \)