# Thread:

1. ## Help with finding volumes of rotated curves

Hey I was recently introduced to the disc, cylindrical shell, and washer methods and I don't understand them fully mostly because it's difficult for me to imagine the shapes created when rotating curves and i dont know which method to use. Here is the problem:

Consider the solid obtained by rotating the region bounded by the given curves about the x-axis.
Find the volume V of this solid.

I decided to use the disc method and set up my integral like so:

$\pi\int(16-4x^2)^2dx$

With -2 and 2 for the limits of integration...solving this yields 64 $\pi$ but i cant get the correct volume.

2. 1) I always try to do it both ways, just to emphasize the concepts in my mind.

2) That is not $64\pi$. Please demonstrate your work.

3) You have it set up exactly correctly. Try the integral again.

4) And the other way: $2\pi \int_{0}^{16}y \left[ 2\sqrt{\frac{16-y}{4}}\right] \;dy$. See if you get the same answer.

5) Please learn to observe an exploit symmetries. The limits of your integral could have been [0,2] with the result multiplied by 2.

3. Thank you for the helpful reply i see what i did, or rather, didn't do.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9456909894943237, "perplexity_flag": "middle"}