Question

Math

Posted 5 months ago

`Calculate the volume of the solid formed by rotating the area between $y = x^2$ and $y = 0$ around the x-axis from $x = 0$ to $x = 1$.`

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To find the volume of the solid formed by rotating the area between $y = x^2$ and $y = 0$ around the x-axis, we use the disk method

step 2

The volume of a thin disk with radius $r$ and thickness $dx$ is $V = \pi r^2 dx$

step 3

Here, the radius of the disk is equal to $y = x^2$. So, the volume of each disk is $V = \pi (x^2)^2 dx = \pi x^4 dx$

step 4

We integrate this expression from $x = 0$ to $x = 1$ to find the total volume: $V = \int_0^1 \pi x^4 dx$

step 5

Using the asksia-ll calculator result, the integral is $\int_0^1 \pi x^4 dx = \frac{\pi}{5}$

Answer

The volume of the solid is $\frac{\pi}{5}$.

Key Concept

Disk Method for Volume

Explanation

The disk method involves slicing the solid into thin disks and summing their volumes using integration. The radius of each disk is given by the function describing the shape of the solid.

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