I A Maron Solutions for Chapter: Applications of the Definite Integral, Exercise 8: Geometrical Applications of the Definite Integral

The axes of two circular cylinders with equal bases intersect at right angles. Compute the surface area of the solid constituting the part common to both cylinders.

Find the areas of the figures bounded by the curves represented parametrically :

$x=2t-t^2; y=2t^2-t^3$

Find the areas of the figures bounded by the curves represented parametrically :

$x=\frac{t^2}{1+t^2}; y=\frac{t\left(1-t^2\right)}{1+t^2}$

Find the area of the figure bounded by the curve given in polar coordinates as $\rho =a \sin 3\phi$ (a three-leaved rose).

Find the areas of the figures bounded by the curves given in polar coordinates :

$\rho =3\sin \phi$ and $\rho =\sqrt{3}\cos \phi$

Find the arc length of the curve $y^2=\frac{4}{9}{\left(2-x\right)}^2$ cut off by the straight line $x=-1$.

Find the volume of the right elliptical cone whose base is an ellipse with semi-axes $a$ and $b$, its altitude being equal to $h$.

Find the volume of the solid generated by revolving about the $x$-axis the figure bounded by the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the straight line $2ay-bx=0$ and the axis of abscissas.