The area of the region bounded by the curves x² + y² = 8 and y² = 2x is

• $2\mathrm{\pi } + \frac{1}{3}$

• $\mathrm{\pi } + \frac{1}{3}$

• $2\mathrm{\pi } + \frac{4}{3}$

• $\mathrm{\pi } + \frac{4}{3}$

The area of the region bounded by the curves, y² = 8x and y = x is

• $\frac{64}{3}$

• $\frac{32}{3}$

• $\frac{16}{3}$

• $\frac{8}{3}$

The area of the region bounded by the curve y = 2x - x² and X - axis is

• $\frac{2}{3} \mathrm{sq} \mathrm{units}$

• $\frac{4}{3} \mathrm{sq} \mathrm{units}$

• $\frac{5}{3} \mathrm{sq} \mathrm{units}$

• $\frac{8}{3} \mathrm{sq} \mathrm{units}$

The area of the region bounded by the lines y = 2x + 1, y = 3x + 1and x = 4 is

• 16 sq unit

• $\frac{121}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{121}{6} \mathrm{sq} \mathrm{unit}$

• 8 sq unit

195.

A closed figure S is bounded by the hyperbola x² - y² = a² and the straight line x = a + h; (h > 0, a > 0). This closed figure is rotated about the X-axis. Then, the volume of the solid ofrevolution is

• ${\mathrm{\pi h}}^2\left(3\mathrm{a} + \mathrm{h}\right)$

• $\frac{{\mathrm{\pi h}}^2}{6}\left(3\mathrm{a} + \mathrm{h}\right)$

• $\frac{{\mathrm{\pi h}}^2}{3}\left(3\mathrm{a} + \mathrm{h}\right)$

• $\frac{{\mathrm{\pi h}}^2}{2}\left(3\mathrm{a} + \mathrm{h}\right)$

The centre of gravity (centre of mass) of a rod (of length L), whose linear mass density varies as the square of the distance from one end is at

• $\frac{3\mathrm{L}}{5}$

• $\frac{2\mathrm{L}}{5}$

• $\frac{\mathrm{L}}{3}$

• $\frac{3\mathrm{L}}{4}$

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

• $2\sqrt{2}\mathrm{\pi }$

• $3\sqrt{2}$

• $3\sqrt{2}\mathrm{\pi }$

• None of these

The area of the portion of the circle x² + y² = 64 which is exterior to the parabola y² = 12x, is

• $\left(8\mathrm{\pi } - \sqrt{3}\right) \mathrm{sq} \mathrm{units}$

• $\frac{16}{3}\left(8 - \sqrt{3}\right) \mathrm{sq} \mathrm{units}$

• $\frac{16}{3}\left(8\mathrm{\pi } - \sqrt{3}\right) \mathrm{sq} \mathrm{units}$

• None of the above

For which of the followmg value of m, is the area of the region bounded by the curve y = x - x² and the line y = mx equals to $\frac{9}{2}$ ?

• - 4

• - 2

• 2

• - 4

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

• $\frac{37\mathrm{\pi }}{3}$

• $7\mathrm{\pi }\sqrt{2}$

• $37\mathrm{\pi }$

• $\frac{7\mathrm{\pi }}{\sqrt{2}}$