Mathematics Integration Level: Misc Level

Sand is pouring from a pipe at the rate of 12 cm3/sec. The falling sand forms a cone on the ground in such a way that the height of is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Mathematics Integration Level: Misc Level

A circular cone with semi-vertical angle 45^{o}, is fixed with its axis vertical an its vertex downwards. Water is poured into the cone at the rate of 2 cm^{3} per second. Find the rate at which the depth of the water id increasing when the depth is 4 cm.

Mathematics Integration Level: Misc Level

A kite is 120 m high and 130 m of string is out. If the kite is moving horizontally at the rate of 5.2 m/sec, find the rate at which the string is being paid out at that instant.

Mathematics Integration Level: Misc Level

A girl of height 1.6 m walks at the rate of 50 meters per minute away from a lamp which is 4 m above the ground. How fast is the girl's shadow lengthening?

Mathematics Integration Level: Misc Level

The top of a ladder 6 meters long is resting against a vertical wall. Suddenly, the ladder begins to slide downwards. At the instant when the foot of the ladder is 4 meters from the wall, it is sliding away at the rate of 0.5 m/sec. How fast is the top sliding downwards at this moment? How far is the foot from the wall when the foot and the top are moving at the same rate?

Mathematics Integration Level: Misc Level

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x=8 cm, find the rate of change of

(i) the perimeter

(ii) the area of the rectangle

Mathematics Integration Level: Misc Level

A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?

Mathematics Integration Level: Misc Level

Find the rate of change of the whole surface of a closed cylinder of radius r and height h with respect to change in radius.

Mathematics Integration Level: Misc Level

Find the rate of change of area of a circle of radius r with regard to its radius. How fast is the area changing with respect to the radius when the radius is 5 cm?

Mathematics Integration Level: Misc Level

Solve the following integral using properties of definite integrals

Mathematics Integration Level: Misc Level

Prove that ==

Mathematics Integration Level: Misc Level

Solve the following integral using properties of definite integrals

Mathematics Integration Level: Misc Level

Solve the following integral using properties of definite integrals

Mathematics Integration Level: Misc Level

Solve the following integral using properties of definite integrals

Mathematics Integration Level: Misc Level

Solve the following integral using properties of definite integrals