Engineering Mat...

Let the function f satisfies \(f\left( x \right) + 2f\left( {\frac{1}{x}} \right) = {x^2},\;x \ne 0\).

If R is the region 0 ≤ x ≤ y ≤ L, then

\(\mathop \int\!\!\!\int \limits_R \left( {{x^2} + {y^2}} \right)dx\;dy\) is

A curve is given by x² + 2y² = 16 is revolved around x axis. The volume of solid generated is

The value of the integral \(\mathop \smallint \limits_0^3 \frac{{2x}}{{{{\left( {1 - {x^2}} \right)}^{2/3}}}}dx\) is

A mouse is running on a string whose equation is given by \(y = \frac{2}{3}{x^{3/2}}\). The mouse moves along the string from point A whose x-coordinate is 0 to point B where x = 3. Find the distance ran by the mouse.

The value of \(\mathop \smallint \limits_0^{\frac{{\rm{\pi }}}{6}} {\cos ^4}3{\rm{\theta }}{\sin ^3}6{\rm{\theta \;d\theta }}\) is

If \(I = \mathop \smallint \limits_0^2 {x^{\left[ {{x^2}} \right]}}dx\), where [x] denotes greatest integer function. Find I upto 2 decimal places.