Question

Math

Posted 4 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 4 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 $V$ of the solid is given by the integral $V = \int_{a}^{b} \pi r^2 dx$, where $r$ is the radius of the disk at a given $x$, which is $y = x^2$ in this case

step 3

Substituting $y = x^2$ into the formula for volume, we get $V = \int_{0}^{1} \pi (x^2)^2 dx = \int_{0}^{1} \pi x^4 dx$

step 4

Using the asksia-ll calculator result, we have $\int_{0}^{1} \pi x^4 dx = \frac{\pi x^5}{5} \Big|_0^1$

step 5

Evaluating the integral from $0$ to $1$, we get $\frac{\pi (1)^5}{5} - \frac{\pi (0)^5}{5} = \frac{\pi}{5}$

Answer

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

Key Concept

Disk Method for Volume

Explanation

The disk method involves integrating the area of circular disks along the axis of revolution to find the volume of the solid formed.

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