Refer to the figure and find the volume v generated by rotating the given region about the specified line. r3 about ab.

V =  

Explanation:

In the given system of coordinates OXY, the region R₃ is bounded by two functions:

y₁ =  (green line)

y₂ = 2x  (blu line)

in the intervals:

0 ≤ x ≤ 1

0 ≤ y ≤ 2

We need to find the volume of this region rotated about the line AB, which is x = 1. In order to do so, we need to change system of coordinates, such as the rotation is about the y-axis, therefore we need to perform a translation:

After the translation R₃ will be bounded by:

y₁ =

y₂ = 2x + 2

in the intervals:

-1 ≤ x ≤ 0

0 ≤ y ≤ 2

At this point, we can use the washer method (see picture attached). The general formula is:

A = π(R² - r²)

where:

A = area

R = outer radius of a washer

r = inner radius of a washer

Since the radii are x-values which vary with the height, represented by the y-values, we need to write the inverse functions:

[Note: I used the curves on the left side of the graph, but you could find the ones representing the right side of the graph and use those]

Now, we can find the function for the area of each washer:

Therefore the volume of the region R₃ will be:

step-by-step explanation:

first, eliminate the parenthesis:

3

x

+

2

−

4

x

second, group like terms:

3

x

−

4

x

+

2

third, combine like terms:

(

3

−

4

)

x

+

2

→

−

1

x

+

2

→

−

x

+

2