Correct, that is exactly what I show in my 3D sketch around 14:10. The base of the semicircles are perpendicular to the y-axis, parallel to the x-axis, and are neither parallel nor perpendicular to y=(1/2)x.
The line x=0 is the same as the y-axis, not the x axis. So the region below the x-axis is included. If the base was instead bounded by y=0, then you would be correct in saying that it doesn’t include the region under the x-axis. Does that make sense?
What calc 2 topic are you referring to? This video does not deal with rotating regions around an axis (no solids of revolution), we are only looking at solids with bases defined by a function and that have cross sections of common shapes we know the area of. If you are looking for solids of revolution, check out my videos on the Disk method, washer method, and shell method. All can be found in the beginning of my calc 2 playlist. Hope this helps!
I was hoping you would cover volume by cross sections. Great job with the 3-d drawings!
Thanks! They can definitely be tricky to draw sometimes.
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For Example 2, The semi circle is perpendicular to the y-axis, which implies that it is parallel to the x axis, not the line y = 1/2 x
Correct, that is exactly what I show in my 3D sketch around 14:10. The base of the semicircles are perpendicular to the y-axis, parallel to the x-axis, and are neither parallel nor perpendicular to y=(1/2)x.
In the last example, the base is bounded by x=0 so isnt that supposed to mean that it wont include the region under the x-axis
The line x=0 is the same as the y-axis, not the x axis. So the region below the x-axis is included. If the base was instead bounded by y=0, then you would be correct in saying that it doesn’t include the region under the x-axis. Does that make sense?
Do you have video that talks about the instances when it is not rotated in non zero axis
What calc 2 topic are you referring to? This video does not deal with rotating regions around an axis (no solids of revolution), we are only looking at solids with bases defined by a function and that have cross sections of common shapes we know the area of. If you are looking for solids of revolution, check out my videos on the Disk method, washer method, and shell method. All can be found in the beginning of my calc 2 playlist. Hope this helps!