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castortr0y [4]
3 years ago
12

College calculus: volume of a solid

Mathematics
1 answer:
Oliga [24]3 years ago
3 0
First, start by graphing the given region.
You should get a parabolic shape with min at (0,6) and end points intersecting with y=7.
When you rotate this around y=5, it will look like a vertical doughnut.
The cross-section of a doughnut is circular and looks like a washer.

Hence, we will use the 'washer method'
V = \pi  \int\limits^a_b {R^2 - r^2} \, dx
R is radius of outer ring from center which is y=5. The max value of region is y=7, giving a distance of 2 for our radius.
R = 2
'r' is radius of inner ring, this is point along curve f(x)=5+sec(x).
The distance from y=5 would be f(x) - 5.
r = sec(x)

V = \pi  \int\limits^{\pi/3}_{-\pi/3} {4 - sec^2 (x)} \, dx  \\  \\  \\ V = \pi |_{-\pi/3}^{\pi/3}(4x - tan x) \\  \\  \\ V = \frac{8\pi^2}{3} - 2\sqrt{3} \pi
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Step-by-step explanation:

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