Question

Region R is bounded by the lines y=4, x=6, the y-axis and the x-axis. Region R is rotated about the line y=-2. Find the volume of Region R after being rotated.

Answer 1

Answer:

The volume of the region after being rotated is 603.186 unit³

Step-by-step explanation:

Here we have the shape formed by the region is that of a hollow cylinder with thickness 4 and length 6 with radius of opening at center = 2

Therefore, the volume is

πR²L - πr²L = πL(R² - r²)

Where:

R = Outer radius of the cylinder = 4 + 2 = 6

r = inner radius = 2

L = Length of the cylinder = 6

Therefore we have

Volume  = π×6×(6² - 2²) = 603.186 unit³.

Answer 2

The volume of rotation of region R is $192\pi$ cubic units.

Given information:

Region R is bounded by the lines y=4, x=6, the y-axis and the x-axis.

Region R is rotated about the line y=-2.

So, the region R is a rectangle with length 6 units (x=0 to x=6), and the height of the rectangle is 4 units (y=0 to y=4).

After rotating this rectangle about y=-2, we will get a hollow cylinder.

The outer radius of the cylinder will be,

$R=4-(-2)=6$

The inner radius of the cylinder will be,

$r=0-(-2)=2$

And the length of the cylinder is 6 units.

So, the volume of the solid formed after the rotation will be,

$V=\pi R^2L-\pi r^2L\\V=\pi L(R^2-r^2)\\V=\pi \times 6(6^2-2^2)\\V=\pi\times 6 \times 32\\V=192\pi$

Therefore, the volume of rotation of region R is $192\pi$ cubic units.

For more details about Volume after rotation, refer to the link:

https://brainly.com/question/1518570