Explained: "Find the volume of the solid generated by..."

djeljenike

Answered question

2021-05-19

Find the volume of the solid generated by revolving the standed region about the x-axis.

Te volume of the solid is ? cubic units.
The equation:

4 x + 3 y = 24

stuth1

Skilled2021-05-20Added 97 answers

Step 1

\Rightarrow y = 8 - (\frac{4}{3}) x

\Rightarrow y = 4 \times (2 - (\frac{1}{3}) x)

for x intercept,

y = 0

0 = 4 \times (2 - (\frac{1}{3}) x)

(\frac{1}{3}) x = 2

x = 6

Volume generated by rotating the given region about x axis by washer method

v = \int_{0}^{6} π [4 \times (2 - (\frac{1}{3}) x)]^{2} d x

v = \int_{0}^{6} 16 π [2^{2} - 2 \times 2 (\frac{1}{3}) x + ((\frac{1}{3}) x)^{2}] d x

v = \int_{0}^{6} 16 π [4 - (\frac{4}{3}) x + (\frac{1}{9}) x^{2}] d x

v = \int_{0}^{6} 16 π [4 x - (\frac{4}{3}) (\frac{1}{2}) x^{2} + (\frac{1}{9}) (\frac{1}{3}) x^{3}]

v = \int_{0}^{6} 16 π [4 x - (\frac{2}{3}) x^{2} + (\frac{1}{27}) x^{3}]

v = 16 π [4 \times 6 - (\frac{2}{3}) 6^{2} + (\frac{1}{27}) 6^{3}] - 16 π [4 \times 0 - (\frac{2}{3}) 0^{2} + (\frac{1}{27}) 0^{3}]

v = 16 π [4 \times 6 - (2 \times \frac{36}{3}) + (\frac{216}{27})]

v = 16 π [24 - 24 + 8]

v = 128 π

Volume

= 128 π = 402

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