Question

Sketch the region enclosed by the lines x=0 x=6 y=2 and y=6. Identify the vertices of the region. Revolve the region around the y-axis. Identify the solid formed by the revolution calculate the volume of the solid. Leave the answer in terms of pi.

Answer 1

The region describes a rectangle, and the vertices are just the intersections of pairwise-chosen lines, e.g. $x=0$ and $y=2$ intersect at (0, 2), $x=0$ and $y=6$ intersect at (0, 6), and so on.

The four vertices are then (0, 2), (0, 6), (6, 2) and (6, 6).

Since the line $x=0$ coincides with the $y$-axis, revolving the region about this axis will generate a cylinder of radius 6 and height 6 - 2 = 4, so you can expect the volume to be

$\pi(6^2)(4)=144\pi$

We can compute a volume integral to verify this: using the shell method, we have

$\displaystyle2\pi\int_0^64x\,\mathrm dx=4\pi x^2\bigg|_{x=0}^{x=6}=144\pi$

Answer 2

By revolving the region around the y-axis would form a cylinder and its volume is equal to 144π.

How to identify the vertices of the region?

We can deduce that the vertices are intersections of pairwise-chosen lines, such that;

x = 0 and y = 2 would intersect at (0, 2).

x = 0 and y = 6 would intersect at (0, 6).

x = 6 and y = 2 would intersect at (6, 2).

x = 6 and y = 6 would intersect at (6, 6).

Thus, the four vertices are (0, 2), (0, 6), (6, 2) and (6, 6) and the region enclosed by the lines describes a rectangle.

Revolving the region about where x = 0 intersects with the y-axis would form a cylinder of radius (6 units) with height (6 - 2 = 4 units).

Mathematically, the volume of this cylinder is given by this formula:

Volume = πr²h

Where:

• h is the height.

• r is the radius.

Substituting the given parameters into the formula, we have;

Volume = π × 6² × 4

Volume = π × 36 × 4

Volume = 144π.

Read more on volume of a cylinder here: brainly.com/question/21367171

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