Computing Areas

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$, compute the area bounded by the curves $y=x^{2}$, $x=y^{2}$.

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$, compute the area bounded by the curves $y=2x-x^{2}$, $y=x^{2}$.

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$, compute the area bounded by the curves $2y=x^{2}$, $x=y$.

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$, compute the area bounded by the curves $4y=x^{2}-4x$, $x-y-3=0$.

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$ and passing to polar coordinates, compute the area bounded by the lemniscate $(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})$ and the circle $x^{2}+y^{2}=2ax$.

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$ and passing to polar coordinates, compute the area bounded by the cardioid $(x^{2}+y^{2}-ax)^{2}=a^{2}(x^{2}+y^{2})$ and the circle $x^{2}+y^{2}=ay\sqrt{3}$.

Introducing polar coordinates, compute the area bounded by the lemniscate $(x^{2}+y^{2})^{2}=2a^{2}xy$.

Introducing polar coordinates, compute the area bounded by the pedal curve of the ellipse $(x^{2}+y^{2})^{2}=a^{2}x^{2}+b^{2}y^{2}$.

Introducing polar coordinates, compute the area bounded by the curve $(x^{2}+y^{2})^{3}=a^{2}(x^{4}+y^{4})$.

Introducing polar coordinates, compute the area bounded by the curve $(x^{2}+y^{2})^{2}=a^{2}x^{2}-b^{2}y^{2}$.

Introducing polar coordinates, compute the area of the loop of the folium of Descartes $x^{3}+y^{3}=axy$.

Introducing polar coordinates, compute the area bounded by the curve $(x^{2}+y^{2})^{2}=a(x^{3}+y^{3})$.

Introducing polar coordinates, compute the area bounded by the curve $x^{4}+y^{4}=2a^{2}xy$.

Let $\omega (x,y)$, $\varphi (x,y)$, $\psi (x,y)$ be homogeneous functions of degrees $n$, $n-2$, $n-2$, which vanish simultaneously only at $x=0$ and $y=0$ and are positive for other values of $x$ and $y$. Denote by $S(a,b)$ the area bounded by the curve

Prove that $S(a+\alpha , b+\beta )=S(a,b)+S(\alpha ,\beta )$.

In particular, if $\omega (x,y)=(x^{2}+y^{2})^{2}$, $\varphi (x,y)=x^{2}$, $\psi (x,y)=y^{2}$, then $S(a,b)=S(b,a)$. Deduce from this that $S(a,b)=\dfrac {\pi }{2}(a^{2}+b^{2})$ (see problem 3510: here $S(a,b)$ is the area bounded by the curve $\omega (x,y)=a\varphi (x,y)+b\psi (x,y)$ for homogeneous functions $\omega$, $\varphi$, $\psi$ of degrees $n$, $n-2$, $n-2$, which vanish simultaneously only at $x=0$, $y=0$ and are positive at all other points, with $a \geqslant 0$, $b \geqslant 0$).

In the following problems it is useful to apply generalized polar coordinates, setting $x=ar\cos^{\sigma }\varphi$, $y=br\sin^{\sigma }\varphi$, where $a$, $b$, and $\sigma$ are suitably chosen constants; the Jacobian of this substitution equals $J=ab\sigma r\cos^{\sigma -1}\varphi \sin^{\sigma -1}\varphi$.

Find the area bounded by the curve

Find the area bounded by the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}=\dfrac {x}{a}-\dfrac {y}{b}$; $y>0$.

Find the area bounded by the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{3}=\dfrac {x^{2}}{h^{2}}$; $y>0$, $a>0$, $b>0$.

Find the area of the loop of the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{3}=\dfrac {xy}{c^{2}}$.

Find the area bounded by the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {x^{2}}{h^{2}}+\dfrac {y^{2}}{k^{2}}$; $x>0$, $y>0$.

Find the area bounded by the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {x^{2}}{h^{2}}-\dfrac {y^{2}}{k^{2}}$; $x>0$, $y>0$.

Find the area bounded by the curve $\left(\left(\dfrac {x}{a}\right)^{2/3}+\left(\dfrac {y}{b}\right)^{2/3}\right)^{6}=\dfrac {x^{2}}{h^{2}}+\dfrac {y^{2}}{k^{2}}$.

Find the area bounded by the curve $\left(\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}\right)^{8}=\dfrac {x^{2}}{h^{2}}+\dfrac {y^{2}}{k^{2}}$; $x>0$, $y>0$.

Find the area of the loop of the curve $\left(\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}\right)^{8}=\dfrac {xy}{c^{2}}$.

In the following problems the boundary of the area is given by equations of the form $\varphi (x,y,a)=0$, $\varphi (x,y,b)=0$, $\psi (x,y,\alpha )=0$, $\psi (x,y,\beta )=0$. In these it is convenient to express $x$ and $y$ in terms of new variables $u$ and $v$ from the equations $\varphi (x,y,u)=0$, $\psi (x,y,v)=0$. Having found the Jacobian $J$, we can compute $S$ by the formula $\displaystyle S=\iint \limits_{S} dx \, dy=\int \limits_{a}^{b} du \int \limits_{\alpha }^{\beta } \left|J\right| \, dv$.

Find the area bounded by the curves $x+y=a$, $x+y=b$, $y=\alpha x$, $y=\beta x$; $a<b$, $\alpha <\beta$.

Find the area bounded by the curves $x^{2}=ay$, $x^{2}=by$, $y=m$, $y=n$; $a<b$, $m<n$.

Find the area bounded by the curves $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}=1$, $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}=2$, $\dfrac {x}{a}=\dfrac {y}{b}$, $\dfrac {x}{a}=9\dfrac {y}{b}$; $x>0$, $y>0$.

Find the area bounded by the curves $xy=a^{2}$, $xy=b^{2}$, $y=m$, $y=n$; $a^{2}>b^{2}$, $m>n$.

Find the area bounded by the curves $y^{2}=mx$, $y^{2}=nx$, $x=\alpha y$, $x=\beta y$; $m>n$, $\alpha >\beta$.

Find the area bounded by the curves $xy=a^{2}$, $xy=b^{2}$, $x=\alpha y$, $x=\beta y$; $x>0$, $y>0$, $a>b$, $\alpha >\beta$.

Find the area bounded by the curves $y^{2}=ax$, $y^{2}=bx$, $x^{2}=my$, $x^{2}=ny$; $a>b$, $m>n$.

Find the area bounded by the curves $xy=a^{2}$, $xy=b^{2}$, $y^{2}=mx$, $y^{2}=nx$; $a^{2}>b^{2}$, $m>n$.

Find the area bounded by the curves $y^{2}=a^{2}-2ax$, $y^{2}=b^{2}-2bx$, $y^{2}=m^{2}+2mx$, $y^{2}=n^{2}+2nx$; $y>0$, $a>b$, $m>n$.

Find the area bounded by the curves

Find the area bounded by the ellipse $(ax+by+c)^{2}+(a_{1}x+b_{1}y+c_{1})^{2}=h^{2}$; $ab_{1}-a_{1}b \neq 0$.

Find the area bounded by the line $\left|ax+by+c\right|+\left|a_{1}x+b_{1}y+c_{1}\right|=h$; $ab_{1}-a_{1}b \neq 0$.