Evaluate \(\int_C x^4\,\mathrm{d}x + xy\,\mathrm{d}y,\)
where \(C\) is the triangular curve consisting of
the line segments from \((0,0)\) to \((1,0)\)
and from \((1,0)\) to \((0,1)\) and from \((0,1)\) to \((0,0).\)
Stewart
Evaluate \(\oint \Big(3y-\mathrm{e}^{\sin(x)}\Big)\,\mathrm{d}x + \Big(7x+\sqrt{y^4+1}\Big)\,\mathrm{d}y,\)
where \(C\) is the circle \(x^2+y^2=9.\)
Stewart
Calculate the area enclosed by the ellipse
\(\frac{1}{a^2}x^2+\frac{1}{b^2}y^2=1.\)
Stewart
Evaluate \(\oint y^2\,\mathrm{d}x + 3xy\,\mathrm{d}y,\)
where \(C\) is the boundary of the semiannular region \(R\)
in the upper half-plane between the circles
\(x^2+y^2=1\) and \(x^2+y^2=4.\)