A circular plate of radius r feet is submerged vertically in a tank of fluid that weighs w pounds per cubic foot. The center of the circle is k feet below the surface of the fluid, where k > r. Show that the fluid force on the surface of the plate is F = wk(πr²). (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)

Calculate the force on one side of a circular plate with radius 2 m, submerged vertically in a tank of water so that the top of the circle is tangent to the water surface.

A semicircular plate of radius r meters, oriented, is submerged in water so that its diameter is located at a depth of m meters. Calculate the fluid force on one side of the plate in terms of m and r.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Rectangle: 4 feet across the top and with a height of 3 feet.

Alyssa and Gabriel play the following game. Alyssa rolls a die. If she rolls a 1, 2, or 3. Gabriel gives Alyssa $3. If Alyssa rolls a 4 or 5. Gabriel gives Alyssa$2. However, if Alyssa rolls a 6, she gives Gabriel $14. (a) Determine Alyssas expectation. (b) Determine Gabriels expectation.

Use the Theorem of Pappus to find the volume of the given solid. A cone with height h and base radius r.

On August 12, 2000, the Russian submarine Kursk sank to the bottom of the sea, 350 feet below the surface. Find the following at the depth of the Kursk. (a) The water pressure in pounds per square foot and pounds per square inch. (b) The force on a 5-foot square metal sheet held (i) Horizontally. (ii) Vertically.

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank when the tank is half full, where the diameter is 3 feet and the gasoline weighs 42 pounds per cubic foot.

Find the unit tangent vector T(t ) at the point with the given value of the parameter t. r(t) = (t2 - 2t,1 + 3t, 1/3t3 + 1/2t2), t = 2

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. (The weight-density of water is 62.4 pounds per cubic foot. Round your answer to two decimal places.) Parabola, y = x^2.

Consider two particles of masses $m_{1}$ and $m_{2}$. The position of the first particle is fixed, and the distance between the particles is a units. Using Newton’s Law of Universal Gravitation, find the work needed to move the second particle so that the distance between the particles increases to b units.

Find the mass of the two dimensional object that is centered at the origin. A jar lid of radius 3 in. with density function $\rho(x)=\ln (x+1)$

Find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meter. Given: 1 meter across the bottom, with a height of 5 meters and 1 meter below the surface of water

An overhead garage door has two springs, one on each side of the door. A force of 15 pounds is required to stretch each spring 1 foot. Because of the pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the work done by the pair of springs.