The part of straight line y = x + 1 between x = 2 and x = 3 is revolved about x-axis, then the curved surface of the solid thus generated is

• $\frac{37\mathrm{\pi }}{3}$

• $7\mathrm{\pi }\sqrt{2}$

• $37\mathrm{\pi }$

• $\frac{7\mathrm{\pi }}{\sqrt{2}}$

The part of circle x² + y² = 9 in between y = 0 and y = 2 is revolved about y-axis. The volume of generating solid will be

• $\frac{46}{3}\mathrm{\pi }$

• $12\mathrm{\pi }$

• $16\mathrm{\pi }$

• None of these

For $0 \le \mathrm{x} \le \mathrm{\pi }$, the area between the curve y = sin(x) and x - axis is

• 1 sq unit

• 0 sq unit

• 2 sq unit

• - 1 sq unit

The area bounded by the curve y²(2a - x) = x³ and the line x = 2a is

• $3{\mathrm{\pi a}}^2$ sq unit

• $\frac{3{\mathrm{\pi a}}^2}{2} \mathrm{sq} \mathrm{unit}$

• $\frac{3{\mathrm{\pi a}}^2}{4} \mathrm{sq} \mathrm{unit}$

• $\frac{6{\mathrm{\pi a}}^2}{5} \mathrm{sq} \mathrm{unit}$

The area bounded by the curves y² = 4a²(x - 1) and lines x = 1 and y = 4a is

• 4a² sq unit

• $\frac{16\mathrm{a}}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{16{\mathrm{a}}^2}{3} \mathrm{sq} \mathrm{unit}$

• None of these

The solution of the differential equation ${\left(\mathrm{x} + \mathrm{y}\right)}^2\frac{\mathrm{dy}}{\mathrm{dx}} = {\mathrm{a}}^2$ is

• ${\left(\mathrm{x} + \mathrm{y}\right)}^2 = \frac{{\mathrm{a}}^2\mathrm{x}}{2} + \mathrm{C}$

• (x + y)² = a²x + C

• (x + y)² = 2a²x + C

• None of the above

The area bounded by y = log(x), x-axis and ordinates x = 1, x = 2 is

• $\frac{1}{2}{\left(\log \left(2\right)\right)}^2$

• log(2/e)

• log(4/e)

• log(4)

The area of the segment of a circle of radius a subtending an angle of 2$\mathrm{\alpha }$ at the centre is :

• ${\mathrm{a}}^2\left(\mathrm{\alpha } + \frac{1}{2}\sin \left(2\mathrm{\alpha }\right)\right)$

• $\frac{1}{2}{\mathrm{a}}^2\sin \left(2\mathrm{\alpha }\right)$

• ${\mathrm{a}}^2\left(\mathrm{\alpha } - \frac{1}{2}\sin \left(2\mathrm{\alpha }\right)\right)$

• ${\mathrm{a}}^2\mathrm{\alpha }$

109.

Area bounded by the curve y = log_e(x), x = 0, y $\le$ 0 and x - axis is :

• 1 sq unit

• $\frac{1}{2}$ sq unit

• 2 sq unit

• None of these

The volume of the solid generated by the revolution of the curve y = $\frac{{\mathrm{a}}^3}{{\mathrm{a}}^2 + {\mathrm{x}}^2}$ about x - axis is :

• $\frac{1}{2}{\mathrm{\pi }}^3{\mathrm{a}}^2$

• ${\mathrm{\pi }}^3{\mathrm{a}}^2$

• $\frac{1}{2}{\mathrm{\pi }}^2{\mathrm{a}}^3$

• ${\mathrm{\pi }}^2{\mathrm{a}}^3$