Area below x axis not considered when finding volume of shape rotated around y axis?While practicing finding the volume of shapes bounded by a function and the x-axis by rotating the shape around the y-axis and taking an integral (the "washer" or "pipe" method) I became confused as to why the shape used to calculate volume "stops" at the x axis, and isn't a part of the calculation.To explain further, if the shape is being discretized in the x direction (dx), then the integral will be set up to be calculated between two x values, call them a and b. This restricts the calculation to be determining the area only between the x values of a and b. In the y direction, however, there is no such restriction, and so far I have been assuming that the y component of the calculation "begin" at the function being integrated and "end" at the x-axis. I do not understand, however, the mathematics behind why I get the correct answers for the volume of the rotated shape if I am ignoring the area that extends from the x axis to negative infinity in the y direction.A simple example that can help illustrate my question is finding the volume of the shape defined by y = x 2 between 0 &lt; x &lt; 2 rotated around the y axis. Why, in this example, is the area below the x axis not considered?

Question

Area below x axis not considered when finding volume of shape rotated around y axis?While practicing finding the volume of shapes bounded by a function and the x-axis by rotating the shape around the y-axis and taking an integral (the &quot;washer&quot; or &quot;pipe&quot; method) I became confused as to why the shape used to calculate volume &quot;stops&quot; at the x axis, and isn&#039;t a part of the calculation.To explain further, if the shape is being discretized in the x direction (dx), then the integral will be set up to be calculated between two x values, call them a and b. This restricts the calculation to be determining the area only between the x values of a and b. In the y direction, however, there is no such restriction, and so far I have been assuming that the y component of the calculation &quot;begin&quot; at the function being integrated and &quot;end&quot; at the x-axis. I do not understand, however, the mathematics behind why I get the correct answers for the volume of the rotated shape if I am ignoring the area that extends from the x axis to negative infinity in the y direction.A simple example that can help illustrate my question is finding the volume of the shape defined by   y  =      x    2   between   0  &amp;lt;  x  &amp;lt;  2 rotated around the y axis. Why, in this example, is the area below the x axis not considered?

Kamden Simmons · Accepted Answer

Explanation:In your example there is no area below the x axis, but if your function were   y  =  x  −  1 from   x  =  0 to   x  =  2, rotated around the y axis, you are considering a cone and definitely want the area below the x axis

Area below x axis not considered when finding volume of shape rotated around y axis?

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