6.6 Integral Applications to Physics & Engineering

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The tank in the image below is full of water. Determine the Reimann Sum that defines the work required to pump the water out of the spout at the top. On the image below label x_i and delta x. Evaluate the integral generated from the sum.

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The tank in the image below is full of water. Determine the Reimann Sum that defines the work required to pump the water out of the spout at the top. On the image below label x_i and delta x. Evaluate the integral generated from the sum.

10

A cable that weighs 5 kg/m is used to lift 1600kg of coal up a mine shaft that is 250m deep. Draw a picture, label x_i and delta x. Then determine the Reimann Sum that defines the work. Evaluate the integral to determine the total work done.

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A 3.5m chain weighs 12N and hangs from the ceiling. Find the work done in lifting the lower end of the chain to ceiling so that it is level with the upper end(The final shape is a U). Draw a picture, label x_i and delta x. Then determine the Reimann Sum that defines the work. Evaluate the integral to determine the total work done.

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A bucket weighs 3N and a rope of negligible weight are used to draw water from a well 80 feet below. The bucket is filled with 20N of water and is pulled up at a rate of 2 ft/s, but water leaks out a hole in the bucket, dear Liza, at a rate of 0.2 N/s. Find the work done in pulling the bucket to the top of the well. Draw a picture, label x_i and delta x. Then determine the Reimann Sum that defines the work. Evaluate the integral to determine the total work done.

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A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density 820 kg/m^3 to a depth of 1.5 m. Find (a) the hydrostatic pressure on the bottom of the tank, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the tank. Draw a picture, label x_i and delta x. Then determine the Reimann Sum that defines the pressure. Evaluate the integral to determine the total work done.

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A vertical plate is partially submerged in water and has the indicated shape. Determine the hydrostatic force against the face of the plate. On the picture below be sure to label any necessary r values or x_i and delta x. Then determine the Reimann Sum that defines the Force. Evaluate the integral to determine the Force.

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A vertical dam has a semi-circular gate at the bottom as shown below. Find the hydrostatic force against the gate. On the picture below be sure to label any necessary r values or x_i and delta x. Then determine the Reimann Sum that defines the Force. Evaluate the integral to determine the force.

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The area enclosed by the following curves:

is shape of the ends of large tank with units in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft with gasoline. (Assume the gasoline’s density is 42 lb/ft^3) Draw a picture below and label any necessary r values or x_i and delta x. Then determine the Reimann Sum that defines the pressure. Evaluate the integral to determine the total pressure.

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A vertical, irregularly shaped plate is submerged in water. The table below shows measurements of its width, taken at the indicated depths. Generate the Reimann Sum and Integral that defines the pressure against the plate. Then use Simpson’s Rule to estimate the force of the water against the plate.

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Three point masses are placed on the x-axis as shown below. Find the moment of the system and center of mass.

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The masses m_i are located at the points P_i . Find the moments M_x and M_y and the center of mass of the system.

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Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

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Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

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Calculate the moments and and the center of mass of a lamina with the a density of 10 cubic units and the shape below.

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6.6 Integral Applications to Physics & Engineering

The tank in the image below is full of water. Determine the Reimann Sum that defines the work required to pump the water out of the spout at the top. On the image below label x_i and delta x. Evaluate the integral generated from the sum.

The tank in the image below is full of water. Determine the Reimann Sum that defines the work required to pump the water out of the spout at the top. On the image below label x_i and delta x. Evaluate the integral generated from the sum.

A cable that weighs 5 kg/m is used to lift 1600kg of coal up a mine shaft that is 250m deep. Draw a picture, label x_i and delta x. Then determine the Reimann Sum that defines the work. Evaluate the integral to determine the total work done.

A vertical dam has a semi-circular gate at the bottom as shown below. Find the hydrostatic force against the gate. On the picture below be sure to label any necessary r values or x_i and delta x. Then determine the Reimann Sum that defines the Force. Evaluate the integral to determine the force.

Three point masses are placed on the x-axis as shown below. Find the moment of the system and center of mass.

The masses m_i are located at the points P_i . Find the moments M_x and M_y and the center of mass of the system.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

Calculate the moments and and the center of mass of a lamina with the a density of 10 cubic units and the shape below.

Calculate the moments and and the center of mass of a lamina with the a density of 2 cubic units and the shape below.

The tank in the image below is full of water. Determine the Reimann Sum that defines the work required to pump the water out of the spout at the top. On the image below label x_i and delta x. Evaluate the integral generated from the sum.

The tank in the image below is full of water. Determine the Reimann Sum that defines the work required to pump the water out of the spout at the top. On the image below label x_i and delta x. Evaluate the integral generated from the sum.

A cable that weighs 5 kg/m is used to lift 1600kg of coal up a mine shaft that is 250m deep. Draw a picture, label x_i and delta x. Then determine the Reimann Sum that defines the work. Evaluate the integral to determine the total work done.

A vertical dam has a semi-circular gate at the bottom as shown below. Find the hydrostatic force against the gate. On the picture below be sure to label any necessary r values or x_i and delta x. Then determine the Reimann Sum that defines the Force. Evaluate the integral to determine the force.

Three point masses are placed on the x-axis as shown below. Find the moment of the system and center of mass.

The masses m_i are located at the points P_i . Find the moments M_x and M_y and the center of mass of the system.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

Calculate the moments and and the center of mass of a lamina with the a density of 10 cubic units and the shape below.

Calculate the moments and and the center of mass of a lamina with the a density of 2 cubic units and the shape below.