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Elden [556K]
3 years ago
6

Evaluate the line integral 2 + x2y ds where c is the upper half of the circle x2 + y2 = 1.

Mathematics
1 answer:
goldfiish [28.3K]3 years ago
8 0

Parameterize <em>C</em> by

r(t) = 〈x(t), y(t)〉 = 〈cos(t), sin(t)〉

with 0 ≤ t ≤ <em>π</em>. Then the line integral is

\displaystyle \int_C (2+x^2y)\,\mathrm ds = \int_0^\pi (2+\cos^2(t)\sin(t))\left\|\mathbf r'(t)\right\|\,\mathrm dt \\\\ = \int_0^\pi (2+\cos^2(t)\sin(t)) \,\mathrm dt = \boxed{\frac23+2\pi}

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