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:

a. turk and mcalister

b. howard, mission, and market

c. yes. since streets end some do not cross yet are not parallel to others. example: mcalister and 5th

step-by-step explanation:

parallel lines are lines which do not intersect ever and look a lot like railroad tracks. turk, mcalister and golden gate are examples of these.

intersect are streets which cross each other and a car on one can reach the other. howard, mission, and market intersect with 6th.

skew lines are lines which neither intersect nor are parallel. these lines exist here because some streets end and never intersect another like mcalister and 5th.

x=6 and x=3

beep beep imma sheep