$2\left(r^2\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+1\right)$$\stackrel{\text{assign to a name}}{\to }$$Q$

Invoke the following task template 

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Polar

Form and evaluate the integral from first principles

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}r Q \ⅆr \ⅆ\mathrm{\θ}$ = ${\∫}_0^{2\mathrm{\π}}{\∫}_0^{1\+\cos \left(\mathrm{\θ}\right)}\left(2r^2\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+2\right)r\ⅆr\ⅆ\mathrm{\θ}$$\=$$3\mathrm{\π}$

Table 9.10.8(a)   Left-hand side of Stokes-form of Green's theorem implemented in polar coordinates

Initialize

Loading Student:-VectorCalculus

Define the Cartesian vector field F

$⟨x-2 y\,x^{2}y⟩$ = $\stackrel{\text{to Vector Field}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$F$$$

Obtain the line integral of F around the cardioid

$\mathbf{F}$ = $\stackrel{\text{line integral}}{\to }$${\∫}_0^{2\mathrm{\π}}\left(-\left(\left(1\+\cos \left(t\right)\right)\cos \left(t\right)-2\left(1\+\cos \left(t\right)\right)\sin \left(t\right)\right)\sin \left(t\right)\left(2\cos \left(t\right)\+1\right)\+{\left(1\+\cos \left(t\right)\right)}^3{\cos \left(t\right)}^2\sin \left(t\right)\left(2{\cos \left(t\right)}^2\+\cos \left(t\right)-1\right)\right)\ⅆt$$\=$$3\mathrm{\π}$$$

Table 9.10.8(b)   Line integral of F around cardioid

$Q≔2\left(r^2\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+1\right)\:$

Use the top-level Int and int commands to integrate in polar coordinates

$\mathrm{Int}\left(r Q\,\left[r\=0..1\+\cos \left(\mathrm{\θ}\right)\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)\=:-\mathrm{int}\left(r Q\,\left[r\=0..1\+\cos \left(\mathrm{\theta }\right)\,\mathrm{\theta }\=0..2 \mathrm{\pi }\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{1\+\cos \left(\mathrm{\θ}\right)}r\left(2r^2\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\θ}\right)\+2\right)\ⅆr\ⅆ\mathrm{\θ}\=3\mathrm{\π}$

Table 9.10.8(c)   Left-hand side of Stokes-form of Green's theorem implemented in polar coordinates

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\:$

$\mathbf{F}≔\mathrm{VectorField}\left(⟨x-2 y\,x^{2}y⟩\right)\:$

Use the LineInt command to form and evaluate ${∳}_{C}f \mathrm{dx}\+g \mathrm{dy}$ 

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨1\+\cos \left(\mathrm{\theta }\right)\,\mathrm{\theta }⟩\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coords}\=\mathrm{polar}\left[r\,\mathrm{\theta }\right]\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{2\mathrm{\π}}\left(-\left(\left(1\+\cos \left(\mathrm{\θ}\right)\right)\cos \left(\mathrm{\θ}\right)-2\left(1\+\cos \left(\mathrm{\θ}\right)\right)\sin \left(\mathrm{\θ}\right)\right)\sin \left(\mathrm{\θ}\right)\left(2\cos \left(\mathrm{\θ}\right)\+1\right)\+{\left(1\+\cos \left(\mathrm{\θ}\right)\right)}^3{\cos \left(\mathrm{\θ}\right)}^2\sin \left(\mathrm{\θ}\right)\left(2{\cos \left(\mathrm{\θ}\right)}^2\+\cos \left(\mathrm{\θ}\right)-1\right)\right)\ⅆ\mathrm{\θ}$

$\mathrm{LineInt}\left(\mathbf{F}\,\mathrm{Path}\left(⟨1\+\cos \left(\mathrm{\theta }\right)\,\mathrm{\theta }⟩\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coords}\=\mathrm{polar}\left[r\,\mathrm{\theta }\right]\right)\right)$ = $3\mathrm{\π}$$$

Table 9.10.8(d)   Line integral of F around the cardioid $r\=1\+\cos \left(\mathrm{\θ}\right)$