It seems you've uploaded an image with mathematical content. Let me analyze it and provide a solution or explanation. Please give me a moment. PIL import Image

Load and display the image to examine its content

image_path = "/mnt/data/1733589280203302bltpa.jpg" image = Image.open(image_path) image.show()It seems I need to extract the content from the image to help you further. Let me process the image to analyze the mathematical content. pytesseract

Use Tesseract to extract text from the image

extracted_text = pytesseract.image_to_string(image) extracted_textThe image contains a question about calculating the volume of a solid obtained by rotating a region bounded by $y = x^2$ and $y = 4$ about the y-axis.

The question asks to identify the correct integral expression to compute this volume.

Here are the two options given:

1. $\int_0^2 \pi x^2 \, dx$
2. $\int_0^4 \pi y^2 \, dy$

To solve this:

Step 1: Identify the Bounds

The region is bounded by $y = x^2$ and $y = 4$. This suggests the upper boundary is $y = 4$, and the curve is described by $y = x^2$.

Step 2: Use the Method of Cylindrical Shells

Since the rotation is about the y-axis, we can use the method of cylindrical shells.

For cylindrical shells, the formula for volume is: $V = \int_{a}^{b} 2\pi \cdot \text{(radius)} \cdot \text{(height)} \, dx$ In this case:

• Radius = $x$ (distance from the y-axis)
• Height = $4 - x^2$ (the distance between the curves $y = 4$ and $y = x^2$)

Thus, the integral should be: $V = \int_0^2 2\pi x \cdot (4 - x^2) \, dx$ This simplifies to: $V = 2\pi \int_0^2 x(4 - x^2) \, dx$ This is the appropriate integral to calculate the volume when rotating around the y-axis.

Step 3: Verify the Provided Options

The first option provided is $\int_0^2 \pi x^2 \, dx$, which is not correct for this problem, as it doesn't consider the function $4 - x^2$ for the height of the shell.

The second option $\int_0^4 \pi y^2 \, dy$ represents a formula for rotation around the x-axis, which is not applicable here.

Thus, the correct approach uses the method of cylindrical shells with the bounds $0$ to $2$, and the integrand involves the product of $x$ and $(4 - x^2)$.

Would you like further details or help solving the integral?