Alan works in a gym. One week he recorded the number of people who
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By Green's theorem, the line integral
is equivalent to the double integral
where <em>D</em> is the region bounded by the curve <em>C</em>, provided that this integrand has no singularities anywhere within <em>D</em> or on its boundary.
It's a bit difficult to make out what your integral should say, but I'd hazard a guess of
Then the region <em>D</em> is
<em>D</em> = {(<em>x</em>, <em>y</em>) : 0 ≤ <em>x</em> ≤ 1 and <em>x</em> ² ≤ <em>y</em> ≤ √<em>x</em>}
so the line integral is equal to
which in this case is 7 times the area of <em>D</em>.
The remaining integral is trivial: