Computing Areas (Quadrature of Curves)

Prove that the area $S$ of the curvilinear trapezoid bounded by the $Ox$ axis, the lines $x=a$, $x=b$, and the cubic parabola $y=Ax^{3}+Bx^{2}+Cx+D$ can be expressed by the formula

where $y_{0}$, $y_{1}$, and $y_{2}$ are the ordinates at $x=a$, $\frac{a+b}{2}$, and $b$.

Prove that the area of the curvilinear trapezoid bounded by the $Ox$ axis, the lines $x=-h$, $x=k$, and the parabola passing through the points $(-h,y_{1})$, $(0,y_{2})$, $(k,y_{3})$, can be expressed by the formula

Prove that the area $S$ of the curvilinear trapezoid bounded by the $Ox$ axis, the lines $x=a$, $x=b$, and the parabola $y=Ax^{3}+Bx^{2}+Cx+D$ can be computed by the formula

where $y_{1}$, $y_{2}$, $y_{3}$ are the values of $y$ at $x=\frac{a+2}{2}-\frac{b-a}{2}\frac{1}{\sqrt{2}}$, $x=\frac{a+b}{2}$, $x=\frac{a+b}{2}+\frac{b-a}{2}\frac{1}{\sqrt{2}}$. (Chebyshev.)

Prove that the same area for the parabola of degree 5 $y=Ax^{5}+Bx^{4}+Cx^{3}+Dx^{2}+Ex+F$ can be computed by the formula $S=\frac{b-a}{18}(5y_{1}+8y_{2}+5y_{3})$, where $y_{1}$, $y_{2}$, $y_{3}$ are the values of $y$ at

(Gauss.)

Prove that the area $S$ of a segment of a parabola can be expressed by the formula $S=\frac{2}{3}lh$, where $l$ is the length of the chord of the segment, and $h$ is the height of the segment.

Find the area of the figure bounded by the parabolas $y=x^{2}$, $y^{2}=x$.

Find the area bounded by one arch of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)$ and the $Ox$ axis.

Find the area bounded by the cardioid $r=a(1+\cos \varphi )$.

Find the area between $Oy$ and the curve $y=\pm \sqrt{\frac{1-x}{x}}$.

Find the area of the astroid $x^{2/3}+y^{2/3}=a^{2/3}$, by bringing its equation to parametric form.

Find the area bounded by the curve $x^{4}+y^{4}=x^{2}+y^{2}$, by switching to polar coordinates.

By switching to polar coordinates, find the area bounded by the loop of the folium of Descartes $x^{3}+y^{3}=3axy$.

By switching to polar coordinates, find the area bounded by the lemniscate $(x^{2}+y^{2})^{2}=2a^{2}xy$.

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

By switching to polar coordinates, find the area bounded by the $Ox$ axis and the tractrix

Here it is convenient to introduce a parameter $\varphi$ by setting $y=a\sin \varphi$.

Find the area lying inside the circle $x^{2}+y^{2}=4px$ and the parabola $y^{2}=2px$.

Find the area between the curves $27py^{2}=8(x-p)^{3}$ and $y^{2}=2px$.

Find the area between a parabola and one of its normals.

Draw a normal to the parabola cutting off a segment of least area.

Find the area lying above the $Ox$ axis and bounded by the lines: $y^{2}=4ax$; $x^{2}+y^{2}=2ax$; $2x-y=4a$.

Find the area of the curvilinear quadrilateral bounded by the parabolas: $y^{2}-9ax+81a^{2}=0$; $y^{2}-4ax+16a^{2}=0$; $y^{2}-ax+a^{2}=0$ and the $Ox$ axis. Show that this area equals the area bounded by the chords.

Determine the area lying between the circle $x^{2}+y^{2}=a^{2}$ and its pedal curve, with respect to the interior point $(b,0)$.

The cycloid

is intersected by the line $y=ka$ (with $0<k<2$). Determine the areas bounded by these lines.

The elongated cycloid $x=a(kt-\sin t)$, $y=a-a\cos t$ (with $0<k<1$) forms loops. Determine the area of the loop. (In particular for $k=\frac{2}{\pi }$; $k=\frac{3}{5\pi }$.)

Find the area of the loop of the strophoid $(a-x)y^{2}=(a+x)x^{2}$.

Find the area bounded by the lines: $y(x^{2}+4)=4(2-x)$; $x=0$; $y=0$.

Find the area bounded by the line $(y-x)^{2}=x^{3}$ and the $Ox$ axis.

Find the area bounded by the half-wave of the curve $y=\sin^{3}x+\cos^{3}x$ and the $Ox$ axis.

Determine the area bounded by the parabola

and the coordinate axes.

Determine one of the areas between the ellipse

and the hyperbola $xy=4a^{2}$.

Determine the area between the parabola $ay=(x-3a)^{2}$ and the hyperbola $xy=4a^{2}$.

Determine the area between the $Ox$ axis, the involute of a circle

and its first vertical tangent.

Determine the area of the part of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ lying outside the circle $x^{2}+y^{2}=ab$.

Determine the area of the curvilinear square belonging to both ellipses

Show that the area between one arch of an epicycloid and the base circle equals $\pi b^{2}\left(3+2\frac{b}{a}\right)$, where $a$ and $b$ are respectively the radii of the base circle and the rolling circle.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r=a\cos \varphi$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r=a\cos 2\varphi$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r=a\cos 3\varphi$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r=a\cos 4\varphi$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r^{2}=a^{2}\frac{\sin 3\varphi }{\sin \varphi }$ for $0<\varphi <\frac{\pi }{3}$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: the ellipse $r=\frac{p}{1+e\cos \varphi }$, $e<1$, $0<\varphi <\varphi_{0}$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: the hyperbola $\frac{p}{1+e\cos \varphi }$, $e>1$, $0<\varphi <\varphi_{0}$; $e\cos \varphi_{0}>-1$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: the equilateral hyperbola

By switching to polar coordinates, find the area bounded by the line

for $0<\omega <\omega_{0}$.

By switching to polar coordinates, find the area bounded by the line

for $0<\omega <\omega_{0}$.

The conchoid of Nicomedes $r=\frac{a}{\sin \varphi }\pm b$, for $a<b$, has a loop. Find the area of the loop.

Find the area bounded by the outer contour of the curve

The curve $r=a\left(\frac{\sqrt{3}}{2}+\cos \frac{\varphi }{2}\right)$ has three loops, whose areas are: $a^{2}\left(\frac{5}{4}\pi +2\sqrt{3}\right)$; $a^{2}\left(\frac{5}{6}\pi -\frac{5}{4}\sqrt{3}\right)$; $a^{2}\left(\frac{5}{12}\pi -\frac{3}{4}\sqrt{3}\right)$. Verify this.

Determine the area lying between the loops of the limaçon of Pascal $r=a+b\cos \varphi$, for $a<b$.

Find the area lying between the curves $r^{2}\cos 2\varphi =a^{2}$ and $r^{2}\cos 2\varphi =4a^{2}\cos^{4}\varphi$.

Determine the area of the lune lying inside the lemniscate $r^{2}=a^{2}\cos 2\varphi$ and outside the circle $r=\frac{a}{\sqrt{2}}$.

Determine the areas bounded by the arcs of the cardioid $r=a(1+\cos \varphi )$ and the circle $r=\frac{3}{2}a$.

Determine the area of one loop of the curve

Determine the area bounded by the curve $x^{4}+y^{4}=2a^{2}xy$.

In what ratio does the line $x+y=2a$ divide the area of the loop of the folium of Descartes $x^{3}+y^{3}=3axy$?

In what ratio does the parabola $y^{2}=ax$ divide the area of the loop of the folium of Descartes $x^{3}+y^{3}=3axy$?

Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and the ellipse obtained from it by rotating about the center through an angle $\alpha <\frac{\pi }{2}$. Determine the area occupied by all points lying inside both ellipses.

Determine the area of the lune lying inside one and outside the other ellipse of the previous problem.

Previous problem (3362): given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and the ellipse obtained from it by rotating about the center through an angle $\alpha <\frac{\pi }{2}$.

Determine the area bounded by the loop of the curve $x^{7}+y^{7}=ax^{3}y^{3}$.

Determine the area bounded by the curve