Calculus April 13Volume: the Shell Method. Instead of cutting a cylinder into disks (think of cutting an onion into slices), we will look at layers of.

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Presentation transcript:

Calculus April 13Volume: the Shell Method

Instead of cutting a cylinder into disks (think of cutting an onion into slices), we will look at layers of shells (think of taking the outside layer off an onion, and then the next layer, and then the next layer...

Calculus April 13 Volume: the Shell Method Instead of cutting a cylinder into disks (think of cutting an onion into slices), we will look at layers of shells (think of taking the outside layer off an onion, and then the next layer, and then the next layer... Now, for the shell of a cylinder, if we want the volume of the outermost shell, we would take the total volume and subtract the volume of all the matter inside the shell. In other words, the volume of the cylinder subtract the volume of the hole.

Calculus April 13 Volume: the Shell Method Instead of cutting a cylinder into disks (think of cutting an onion into slices), we will look at layers of shells (think of taking the outside layer off an onion, and then the next layer, and then the next layer... Now, for the shell of a cylinder, if we want the volume of the outermost shell, we would take the total volume and subtract the volume of all the matter inside the shell. In other words, the volume of the cylinder subtract the volume of the hole.

Calculus April 13 Volume: the Shell Method The shell has a width of w.

Calculus April 13 Volume: the Shell Method The shell has a width of w. We need to find the volume of this shell.

Calculus April 13 Volume: the Shell Method The shell has a width of w. We need to find the volume of this shell. To find the volume, we subtract the volume of the cylinders.

Calculus April 13 Volume: the Shell Method The shell has a width of w. We need to find the volume of this shell. To find the volume, we subtract the volume of the cylinders. The radius of the larger cylinder is the outer radius of the shell. The radius of the smaller cylinder is the inner radius of the shell.

Calculus April 13 Volume: the Shell Method The shell has a width of w. We need to find the volume of this shell. To find the volume, we subtract the volume of the cylinders. The radius of the larger cylinder is the outer radius of the shell. The radius of the smaller cylinder is the inner radius of the shell. The average radius of the shell is p.

Calculus April 13 Volume: the Shell Method The shell has a width of w. We need to find the volume of this shell. To find the volume, we subtract the volume of the cylinders. The radius of the larger cylinder is the outer radius of the shell. The radius of the smaller cylinder is the inner radius of the shell. The average radius of the shell is p. Half of width w is on each side of p.

Calculus April 13 Volume: the Shell Method The shell has a width of w. We need to find the volume of this shell. To find the volume, we subtract the volume of the cylinders. The radius of the larger cylinder is the outer radius of the shell. The radius of the smaller cylinder is the inner radius of the shell. The average radius of the shell is p. Half of width w is on each side of p.

Remember, volume of a cylinder is still  r 2 h.

Volume of shell = (volume of cylinder) – (volume of hole)

Remember, volume of a cylinder is still  r 2 h. Volume of shell = (volume of cylinder) – (volume of hole)

Remember, volume of a cylinder is still  r 2 h. Volume of shell = (volume of cylinder) – (volume of hole) Let’s do some algebra, we will need more room.

Remember, since 2p is a coefficient, it may go in front, p(y) or p(x) is the average radius, h(y) or h(x) is the height, and dy or dx is the thickness.

Remember, since 2p is a coefficient, it may go in front, p(y) or p(x) is the average radius, h(y) or h(x) is the height, and dy or dx is the thickness.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. p(x), the radius, is x. h(x), the height, is x – x3. The thickness is dx.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. p(x), the radius, is x. h(x), the height, is x – x3. The thickness is dx.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. Multiply using the distributive property so you can integrate.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. Multiply using the distributive property so you can integrate. Integrate.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. Multiply using the distributive property so you can integrate. Integrate. Plug in the values and go for it.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. Multiply using the distributive property so you can integrate. Integrate. Plug in the values and go for it.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. Multiply using the distributive property so you can integrate. Integrate. Plug in the values and go for it.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. Multiply using the distributive property so you can integrate. Integrate. Plug in the values and go for it.

Example Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x-axis (0 < x < 1) about the y-axis. The axis of revolution is vertical. Multiply using the distributive property so you can integrate. Integrate. Plug in the values and go for it.