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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∻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 from $ x = 0 $ to $ x = 1 $, we use the disk method. ‖ ‖ step 2 ⋮ The volume $ V $ of the solid is given by the integral $ V = \int_{0}^{1} \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 $ V $, we get $ V = \int_{0}^{1} \pi (x^2)^2 dx $. ‖ ‖ step 4 ⋮ Simplifying the integrand, we have $ V = \int_{0}^{1} \pi x^4 dx $. ‖ ‖ step 5 ⋮ According to the asksia-ll calculation list, the integral $ \int_{0}^{1} \pi x^4 dx $ is equal to $ \frac{\pi}{5} $. ‖ ‖ step 6 ⋮ Therefore, the volume $ V $ is $ \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.⚹◊What is the general formula for calculating the volume of a solid formed by rotating an area around the x-axis?⍭ Generate me a similar question◊

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