To find the volume of the solid obtained by rotating the region bounded by the curves about the y-axis, we can use the method of cylindrical shells. The formula for the volume of a cylindrical shell is V = ∫2πrh dx, where r is the radius and h is the height.

Step 1: Identify the radius and height of the cylindrical shells.
The radius of the cylindrical shells is the distance from the y-axis to the curve, which is x. Since we are rotating about the y-axis, x = 25y is the radius. The height of the cylindrical shells is the distance from the x-axis to the curve, which is y.

Step 2: Set up the integral.
The volume V of the solid is given by the integral from 0 to 3 of 2π(25y)y dy.

Step 3: Evaluate the integral.
V = 2π ∫ from 0 to 3 of 25y^2 dy = 2π[25/3 y^3] from 0 to 3 = 2π[25/3 * 27 - 0] = 2π * 225 = 450π cubic units.

So, the volume of the solid obtained by rotating the region bounded by the curves about the y-axis is 450π cubic units.

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To find the volume of the solid obtained by rotating the region bounded by the curves about the y-axis, we can use the method of cylindrical shells. The formula for the volume of a cylindrical shell is V = ∫2πrh dx, where r is the radius and h is the height.

Step 1: Identify the radius and height of the cylindrical shells.
The radius of the cylindrical shells is the distance from the y-axis to the curve, which is x. Since we are rotating about the y-axis, x = 25y is the radius. The height of the cylindrical shells is the distance from the x-axis to the curve, which is y.

Step 2: Set up the integral.
The volume V of the solid is given by the integral from 0 to 3 of 2π(25y)y dy.

Step 3: Evaluate the integral.
V = 2π ∫ from 0 to 3 of 25y^2 dy = 2π[25/3 y^3] from 0 to 3 = 2π[25/3 * 27 - 0] = 2π * 225 = 450π cubic units.

So, the volume of the solid obtained by rotating the region bounded by the curves about the y-axis is 450π cubic units.

Knowee AI · Accepted Answer

To find the volume of the solid obtained by rotating the region bounded by the curves about the y-axis, we can use the method of cylindrical shells. The formula for the volume of a cylindrical shell is V = ∫2πrh dx, where r is the radius and h is the height.

Step 1: Identify the radius and height of the cylindrical shells.
The radius of the cylindrical shells is the distance from the y-axis to the curve, which is x. Since we are rotating about the y-axis, x = 25y is the radius. The height of the cylindrical shells is the distance from the x-axis to the curve, which is y.

Step 2: Set up the integral.
The volume V of the solid is given by the integral from 0 to 3 of 2π(25y)y dy.

Step 3: Evaluate the integral.
V = 2π ∫ from 0 to 3 of 25y^2 dy = 2π[25/3 y^3] from 0 to 3 = 2π[25/3 * 27 - 0] = 2π * 225 = 450π cubic units.

So, the volume of the solid obtained by rotating the region bounded by the curves about the y-axis is 450π cubic units.

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.x = 25y, x = 0, y = 3; about the y-axis

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