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Posted 3 months ago
Calculate the volume of the solid formed by rotating the area between y=x2y = x^2 and y=0y = 0 around the x-axis from x=0x = 0 to x=1x = 1.
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 1
To find the volume of the solid formed by rotating the area between y=x2 y = x^2 and y=0 y = 0 around the x-axis from x=0 x = 0 to x=1 x = 1 , we use the disk method
step 2
The volume V V of the solid is given by the integral V=01πr2dx V = \int_{0}^{1} \pi r^2 dx , where r r is the radius of the disk at a given x x , which is y=x2 y = x^2 in this case
step 3
Substituting y=x2 y = x^2 into the formula for V V , we get V=01π(x2)2dx V = \int_{0}^{1} \pi (x^2)^2 dx
step 4
Simplifying the integrand, we have V=01πx4dx V = \int_{0}^{1} \pi x^4 dx
step 5
According to the asksia-ll calculation list, the integral 01πx4dx \int_{0}^{1} \pi x^4 dx is equal to π5 \frac{\pi}{5}
step 6
Therefore, the volume V V is π5 \frac{\pi}{5}
The volume of the solid is π5 \frac{\pi}{5} cubic units.
Key Concept
Disk Method for Volume
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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