Asksia AI LOGO

Sia

younglife's Question
Math
Posted 7 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.
Sign in to unlock the answer
Answer from Sia
Posted 7 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.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Unlock Smarter Learning with AskSia Super!

Join Super, our all-in-one AI solution that can greatly improve your learning efficiency.

30% higher accuracy than GPT-4o
Entire learning journey support
The most student-friendly features
Study Other Question