Question

Math

Posted 3 months ago

`Calculate the volume of the solid obtained by rotating the region enclosed by $y = x^3$, $y = 0$, $x = 0$, and $x = 2$ 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 $y = x^3$, $y = 0$, $x = 0$, and $x = 2$ around the x-axis, we use the disk method. The volume $V$ is given by the integral: $V = \pi \int_0^2 (x^3)^2 \, dx$

step 2

Simplifying the integral, we have: $V = \pi \int_0^2 x^6 \, dx$

step 3

Evaluating the integral: $V = \pi \left[ \frac{x^7}{7} \right]_0^2 = \pi \left( \frac{2^7}{7} - 0 \right) = \pi \left( \frac{128}{7} \right)$

step 4

Therefore, the volume of the solid is: $V = \frac{128\pi}{7} \approx 57.446$

Answer

The volume of the solid is $\frac{128\pi}{7}$ 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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