Methods of Computing Area

Find the area bounded by the following curves, using the formula $S=\displaystyle \iint_{S} dx\, dy$: $xy=a^{2}$, $x+y=\dfrac {5a}{2}$ $(a>0)$.

Fig. 8.34

Find the area bounded by the following curves, using the formula $S=\displaystyle \iint_{S} dx\, dy$: $y^{2}=2px+p^{2}$, $y^{2}=-2qx+q^{2}$ $(p>0,q>0)$.

Fig. 8.35

Find the area bounded by the following curves, using the formula $S=\displaystyle \iint_{S} dx\, dy$: $(x-y)^{2}+x^{2}=a^{2}$ $(a>0)$.

Fig. 8.36

Transform to polar coordinates to compute the area bounded by the following curves: $(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})$; $x^{2}+y^{2}\geqslant a^{2}$.

Fig. 8.37

Transform to polar coordinates to compute the area bounded by the following curves: $(x^{3}+y^{3})^{2}=x^{2}+y^{2}$; $x\geqslant 0,y\geqslant 0$.

Transform to polar coordinates to compute the area bounded by the following curves: $(x^{2}+y^{2})^{2}=a(x^{3}-3xy^{2})$ $(a>0)$.

Transform to polar coordinates to compute the area bounded by the following curves: $(x^{2}+y^{2})^{2}=8a^{2}xy$; $(x-a)^{2}+(y-a)^{2}\leqslant a^{2}$ $(a>0)$.

Using the formulas $x=ar\cos^{\alpha }\varphi$, $y=br\sin^{\alpha }\varphi$ $(r\geqslant 0)$, introduce generalized polar coordinates (where $a$, $b$, and $\alpha$ are constants chosen in a suitable manner, and $\dfrac {D(x,y)}{D(r,\varphi )}=\alpha ab\, r\cos^{\alpha -1}\varphi \sin^{\alpha -1}\varphi$) to find the area bounded by the following curve (assume the parameters are positive): $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {x}{h}+\dfrac {y}{k}$.

Using the substitution $x=ar\cos \varphi$, $y=br\sin \varphi$ $(r\geqslant 0)$, find the area bounded by the following curves: $\dfrac {x^{3}}{a^{3}}+\dfrac {y^{3}}{b^{3}}=\dfrac {x^{2}}{h^{2}}+\dfrac {y^{2}}{k^{2}}$; $x=0$, $y=0$.

Using the substitution $x=ar\cos \varphi$, $y=br\sin \varphi$ $(r\geqslant 0)$, find the area bounded by the following curves: $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {x^{2}}{h^{2}}+\dfrac {y^{2}}{k^{2}}$ $(x\geqslant 0,y\geqslant 0)$.

Using the substitution $x=ar\cos \varphi$, $y=br\sin \varphi$ $(r\geqslant 0)$, find the area bounded by the following curves: $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {x^{2}}{h^{2}}-\dfrac {y^{2}}{k^{2}}$ $(x\geqslant 0,y\geqslant 0)$.

Using the substitution $x=ar\cos \varphi$, $y=br\sin \varphi$ $(r\geqslant 0)$, find the area bounded by the following curves: $\sqrt[4]{\dfrac {x}{a}}+\sqrt[4]{\dfrac {y}{b}}=1$; $x=0$, $y=0$.

By performing a suitable change of variables, find the area bounded by the following curves: $x+y=a$, $x+y=b$, $y=\alpha x$, $y=\beta x$ $(0<a<b,0<\alpha <\beta )$.

By performing a suitable change of variables, find the area bounded by the following curves: $xy=a^{2}$, $xy=2a^{2}$, $y=x$, $y=2x$ $(x>0;y>0)$.

By performing a suitable change of variables, find the area bounded by the following curves: $y^{2}=2px$, $y^{2}=2qx$, $x^{2}=2ry$, $x^{2}=2sy$ $(0<p<q;0<r<s)$.

By performing a suitable change of variables, find the area bounded by the following curves: $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}=1$, $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}=2$, $\dfrac {x}{a}=-\dfrac {y}{b}$, $\dfrac {4x}{a}=\dfrac {y}{b}$ $(a>0,b>0)$.

By performing a suitable change of variables, find the area bounded by the following curves: $\dfrac {x^{2}}{\lambda }+\dfrac {y^{2}}{\lambda -c^{2}}=1$, where $\lambda$ takes the following values: $\dfrac {1}{3}c^{2}$, $\dfrac {2}{3}c^{2}$, $\dfrac {4}{3}c^{2}$, $\dfrac {5}{3}c^{2}$ $(x\geqslant 0,y\geqslant 0)$.

Find the area bounded by the ellipse $\left(a_{1}x+b_{1}y+c_{1}\right)^{2}+\left(a_{2}x+b_{2}y+c_{2}\right)^{2}=1$ (where $\delta =a_{1}b_{2}-a_{2}b_{1}\neq 0$).

Find the area bounded by the ellipse

and the hyperbola

$(0<u_{1}<u_{2};0<v_{1}<v_{2};x\geqslant 0,y\geqslant 0)$.

Find the area of the cross-section cut from the surface $x^{2}+y^{2}+z^{2}-xy-xz-yz=a^{2}$ by the plane $x+y+z=b$.

Find the area of the cross-section cut from the surface $\dfrac {1}{x}+\dfrac {1}{y}+\dfrac {1}{z}=0$ by the plane $z=1-2(x+y)$.