Problem

The filled rectangle \(0~\le~x~\le~r\) and \(0~\le~y~\le~h\) rotates \(360°\) about \(x~=~0\). Use the diagram to state the generating region, axis, resulting solid, its radius and height, and its volume.

Prompt diagram for M3-037-A11-V01: the generating region, rotation axis, dimensions, and 360-degree sweep with the resulting solid withheld.

Big Picture

What this problem is really about

Read the coordinate bounds relative to the exact rotation axis. For each fixed vertical coordinate, the filled horizontal segment begins on the axis and sweeps a disk whose radius is its perpendicular extent; repeating that disk across the vertical interval sets the solid's height. This disk-stack view identifies the solid, its dimensions, and the matching base-area-times-height volume structure.

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Preview only

The generating region is the filled rectangle, the axis is x = 0, the solid is a right circular cone of radius r and height h, and its volume is (1/3)pi r² h.

The generating region is treated as only its boundary, the axis is x = 0, the result is a hollow cylindrical surface of radius r and height h, and its enclosed volume is 0.

The generating region is the filled rectangle, the axis is x = 0, the solid is a right circular cylinder of radius r and height h, and its volume is pi r² h.

The generating region is the filled rectangle, the axis is x = 0, the solid is a right circular cylinder of radius h and height r, and its volume is pi h² r.

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The filled rectangle \(0~\le~x~\le~r\) and \(0~\le~y~\le~h\) rotates \(360°\) about \(x~=~0\). Use the diagram to state the generating region, axis, resulting solid, its radius and height, and its volume.

What this problem is really about

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