Find x in the given figure if m∠AOC = 124°.
1 answer:
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Answer: 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:
