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ZanzabumX [31]
3 years ago
8

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.? y = 2 + se

c(x), −π/3 ≤ x ≤ π/3, y = 4; about y = 2
Mathematics
1 answer:
sertanlavr [38]3 years ago
7 0

Answer:

The volume of the solid is: \mathbf{V = \pi [ \dfrac{8 \pi}{3} - 2\sqrt{3}]}

Step-by-step explanation:

GIven that :

y = 2 + sec \  x , -\dfrac{\pi}{3} \leq x \leq \dfrac{\pi}{3} \\ \\ y = 4\\ \\ about \  y \ = 2

This implies that the distance between the x-axis and the axis of the rotation = 2 units

The distance between the x-axis and the inner ring is r = (2+sec x) -2

Let R be the outer radius and r be the inner radius

By integration; the volume of the of the solid  can be calculated as follows:

V = \pi \int\limits^{\dfrac{\pi}{3}}_{\dfrac{-\pi}{3}} [(4-2)^2 - (2+ sec \ x -2)^2]dx \\ \\ \\ V = \pi \int\limits^{\dfrac{\pi}{3}}_{\dfrac{-\pi}{3}} [(2)^2 - (sec \ x )^2]dx \\ \\ \\ V = \pi \int\limits^{\dfrac{\pi}{3}}_{\dfrac{-\pi}{3}} [4 - sec^2 \ x ]dx

V = \pi [4x - tan \  x]^{\dfrac{\pi}{3}}_{\dfrac{-\pi}{3}}  \\ \\ \\ V = \pi [4(\dfrac{\pi}{3}) - tan (\dfrac{\pi}{3}) - 4(-\dfrac{\pi}{3})+ tan (-\dfrac{\pi}{3})] \\ \\ \\ V = \pi [4(\dfrac{\pi}{3}) - tan (\dfrac{\pi}{3}) + 4(\dfrac{\pi}{3})- tan (\dfrac{\pi}{3})]  \\ \\ \\ V = \pi [8(\dfrac{\pi}{3})  - 2 \  tan (\dfrac{\pi}{3}) ]

\mathbf{V = \pi [ \dfrac{8 \pi}{3} - 2\sqrt{3}]}

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