Methods of Computing Area

Prove that the area of a parabolic arch equals

where $b$ is the base and $h$ is the height of the arch.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations (throughout this section and all subsequent sections, every parameter is taken to be positive): $ax=y^{2}$, $ay=x^{2}$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $y=x^{2}$, $x+y=2$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $y=2x-x^{2}$, $x+y=0$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $y=\left|\lg x\right|$, $y=0$, $x=0.1$, $x=10$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $y=x$, $y=x+\sin^{2}x$ ($0\leqslant x\leqslant \pi$).

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $\displaystyle y=\dfrac {a^{3}}{a^{2}+x^{2}}$, $y=0$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $\displaystyle \dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $y^{2}=x^{2}(a^{2}-x^{2})$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $y^{2}=2px$, $27py^{2}=8(x-p)^{3}$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $Ax^{2}+2Bxy+Cy^{2}=1$ ($AC-B^{2}>0$).

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $\displaystyle y^{2}=\dfrac {x^{3}}{2a-x}$ (the cissoid), $x=2a$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations:

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $\displaystyle y^{2}=\dfrac {x^{n}}{(1+x^{n+2})^{2}}\quad (x>0;\ n>-2)$.

Find the area enclosed by the curves represented by the following rectangular-coordinate equations: $y=e^{-x}\sin x$, $y=0$ ($x\geqslant 0$).

The parabola $y^{2}=2x$ divides the circle $x^{2}+y^{2}=8$ into two parts. What is the ratio of these two parts' areas?

Express the coordinates of the point $M(x,y)$ on the hyperbola $x^{2}-y^{2}=a^{2}$ as a function of the area $S=OM'M$ of the hyperbolic sector, where this sector is bounded by the hyperbolic arc $M'M$ and the two rays $OM$ and $OM'$, with $M'(x,-y)$ being the point symmetric to $M$ with respect to the $Ox$ axis.

Find the area enclosed by the curve represented by each of the following parametric equations: $x=a(t-\sin t)$, $y=a(1-\cos t)$ ($0\leqslant t\leqslant 2\pi$) (cycloid) and $y=0$.

Find the area enclosed by the curve represented by each of the following parametric equations: $x=2t-t^{2}$, $y=2t^{2}-t^{3}$.

Find the area enclosed by the curve represented by each of the following parametric equations: $x=a(\cos t+t\sin t)$, $y=a(\sin t-t\cos t)$ ($0\leqslant t\leqslant 2\pi$) (involute of a circle) and $x=a$, $y\leqslant 0$.

Find the area enclosed by the curve represented by each of the following parametric equations: $x=a(2\cos t-\cos 2t)$, $y=a(2\sin t-\sin 2t)$.

Find the area enclosed by the curve represented by each of the following parametric equations: $\displaystyle x=\dfrac {c^{2}}{a}\cos^{3}t$, $\displaystyle y=\dfrac {c^{2}}{b}\sin^{3}t$ ($c^{2}=a^{2}-b^{2}$) (evolute of an ellipse).

Find the area $S$ enclosed by the curve represented by each of the following polar equations: $r^{2}=a^{2}\cos 2\varphi$ (lemniscate).

Find the area $S$ enclosed by the curve represented by each of the following polar equations: $r=a(1+\cos \varphi )$ (cardioid).

Find the area $S$ enclosed by the curve represented by each of the following polar equations: $r=a\sin 3\varphi$ (three-petaled rose).

Find the area $S$ enclosed by the curve represented by each of the following polar equations: $\displaystyle r=\dfrac {p}{1-\cos \varphi }$ (parabola), $\varphi =\dfrac {\pi }{4}$, $\varphi =\dfrac {\pi }{2}$.

Find the area $S$ enclosed by the curve represented by each of the following polar equations: $\displaystyle r=\dfrac {p}{1+\varepsilon \cos \varphi }\quad (0\leqslant \varepsilon <1)$ (ellipse).

Find the area $S$ enclosed by the curve represented by each of the following polar equations: $r=a\cos \varphi$, $r=a(\cos \varphi +\sin \varphi )$ ($M\left(-\dfrac {a}{2},0\right)\in S$).

Find the area of the sector enclosed by the curve $\varphi =r\operatorname {arctg}r$ and the two rays $\varphi =0$ and $\displaystyle \varphi =\dfrac {\pi }{\sqrt{3}}$.

Find the area enclosed by the closed curve

Change to polar coordinates to find the area enclosed by each of the following curves: $x^{3}+y^{3}=3axy$ (folium of Descartes).

Change to polar coordinates to find the area enclosed by each of the following curves: $x^{4}+y^{4}=a^{2}(x^{2}+y^{2})$.

Change to polar coordinates to find the area enclosed by each of the following curves: $(x^{2}+y^{2})^{2}=2a^{2}xy$ (lemniscate).

Convert the equation into parametric form to find the area enclosed by each of the following curves: $\displaystyle x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ (astroid).

Convert the equation into parametric form to find the area enclosed by each of the following curves: $x^{4}+y^{4}=ax^{2}y$.