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younglife's Question
Math
Posted 2 months ago
haha
Compute the volume of the solid obtained by rotating the region bounded by y=x2y = x^2, x=0x = 0, and y=4y = 4 about the y-axis.
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
To find the volume of the solid obtained by rotating the region bounded by y=x2y = x^2, x=0x = 0, and y=4y = 4 about the y-axis, we use the method of cylindrical shells
step 2
The height of each cylindrical shell is given by y=x2y = x^2. The radius of each shell is xx
step 3
The volume of a thin shell is given by 2πradiusheightthickness2\pi \cdot \text{radius} \cdot \text{height} \cdot \text{thickness}. Here, the thickness is dxdx
step 4
The volume of the solid is obtained by integrating the volume of the thin shells from x=0x = 0 to x=2x = 2
step 5
Set up the integral: 022πx(x2)dx=2π02x3dx\int_{0}^{2} 2\pi x (x^2) \, dx = 2\pi \int_{0}^{2} x^3 \, dx
step 6
Evaluate the integral: 2π02x3dx=2π[x44]02=2π(244044)=2π(164)=2π4=8π2\pi \int_{0}^{2} x^3 \, dx = 2\pi \left[ \frac{x^4}{4} \right]_{0}^{2} = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right) = 2\pi \left( \frac{16}{4} \right) = 2\pi \cdot 4 = 8\pi
Answer
The volume of the solid is 8π8\pi cubic units.
Key Concept
Cylindrical Shells Method
Explanation
The cylindrical shells method is used to find the volume of a solid of revolution by integrating the volume of thin cylindrical shells.

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