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For a calculus hw assingment I am trying to have a sin wave rotate over an axis, is there any way to do this or input a definite integral into blender? Like the image. enter image description here

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Sure is! Your best bet would be to use geometry nodes. With them you can easily define mathematical expressions and map them onto a curve. Sine function is actually already part of math node so less work for you.

enter image description here

You can then revolve it around the X axis (or whatever axis you want). This can be done either nondestructively using the screw modifier, or destructively using revolve tool (but you have to apply the geometry nodes first). You can also do it using the geometry nodes alone but that is more complicated and it has already been answered here: revolve/screw using GN

enter image description here

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