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barxatty [35]
3 years ago
9

Write and evaluate the definite integral that represents the volume of the solid formed by revolving the region about the x-axis

.
y= -x +4
Mathematics
1 answer:
guapka [62]3 years ago
8 0

Answer: V = \frac{64}{3}\pi

Step-by-step explanation: A solid formed by revolving the region about the x-axis can be considered to have a thin vertical strip with thickness Δx and height y = f(x). The strip creates a circular disk with volume:

V = \pi. y^{2}.Δx

Using the <u>Disc</u> <u>Method</u>, it is possible to calculate all the volume of these strips, giving the volume of the revolved solid:

V = \int\limits^a_b {\pi. y^{2} } \, dx

Then, for the region generated by  y = - x + 4:

V = \int\limits^4_0 {\pi.(-x+4)^{2} } \, dx

V = \pi.\int\limits^4_0 {(x^{2}-8x+16)} \, dx

V = \pi.(\frac{x^{3}}{3}-4x^{2}+16x )

V = \pi.(\frac{4^{3}}{3}-4.4^{2}+16.4 - 0 )

V = \frac{64}{3}.\pi

The volume of the revolved region is V = \frac{64}{3}.\pi

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