PIL import Image import pytesseract

Load the uploaded image

image_path = "/mnt/data/1733292711214siz6hxpj.jpg" extracted_text = pytesseract.image_to_string(Image.open(image_path))

extracted_textThe problem asks for the volume of a solid obtained by rotating the region bounded by $y = 3x^2 - x^3$ and $y = 0$ about the y-axis using the shell method. Here's the solution:

(a) Shell Radius, Circumference, and Height

• Radius: $x$
• Circumference: $2\pi x$
• Height $f(x)$: The height is given by the function $y = 3x^2 - x^3$, so $f(x) = 3x^2 - x^3$.

(b) Shell Method Volume Formula

The volume formula using the shell method is: [ V = 2\pi \int_0^a \text{(radius