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Question
Math
Posted 3 months ago
Calculate the volume of the solid obtained by rotating the region enclosed by y=x3y = x^3, y=0y = 0, x=0x = 0, and x=2x = 2 around the x-axis.
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Answer from Sia
Verified answer
Posted 3 months ago
Solution by Steps
step 1
To find the volume of the solid obtained by rotating the region enclosed by y=x3y = x^3, y=0y = 0, x=0x = 0, and x=2x = 2 around the x-axis, we use the disk method. The volume VV is given by the integral: V=π02(x3)2dxV = \pi \int_0^2 (x^3)^2 \, dx
step 2
Simplifying the integral, we have: V=π02x6dxV = \pi \int_0^2 x^6 \, dx
step 3
Evaluating the integral: V=π[x77]02=π(2770)=π(1287)V = \pi \left[ \frac{x^7}{7} \right]_0^2 = \pi \left( \frac{2^7}{7} - 0 \right) = \pi \left( \frac{128}{7} \right)
step 4
Therefore, the volume of the solid is: V=128π757.446V = \frac{128\pi}{7} \approx 57.446
Answer
The volume of the solid is 128π7\frac{128\pi}{7} or approximately 57.446 cubic units.
Key Concept
The disk method is used to find the volume of solids of revolution by integrating the area of circular disks.
Explanation
The volume is calculated by integrating the area of disks formed by rotating the function around the x-axis, leading to the final result.

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