I'm gonna argue it's muchĮasier to integrate with respect to y here 'cause we already With respect to x, or you could integrate with respect to y. You could say, slices, and then integrate across all of them. Think about the volume of one of these, I guess Many times in integration, what we wanna do is And once again, if we wanted to put, if we wanted to calculate its volume, we could say there'sĪn infinitesimal volume and it would have depth d y. Right over that x y pair, that would sit on that curve. But then our base is theĬorresponding x-value that sits on the curve Here, our y is much lower, it might look some, so our Has an infinitesimal depth, we could think about that And then if we wanted toĬalculate the volume of just a little bit, a slice that X-value that corresponds to that particular y-value. So the base would look like that, it would actually be the So it says the cross section solid taken perpendicular to the y-axis, so let's pick a y-value right over here. And now let's just imagine aĬross section of our solid. So if that's our y-axis and then this is our To visualize the solid and I'll try to do it by drawing this little bit of perspective. Is a rectangle whose base lies in R and whose height is y. Section of the solid taken perpendicular to the y-axis Square root of nine minus x and the axes in the first quadrant. The region enclosed by y is equal to four times the You'll appreciate the number of ways you can do the same problem!! Not required for you right now, but you can come back to this comment if you take up multivariate calc. So, my volume of one slice would be $y^ydxdy$, you'll get the volume over the region as 324, which is exactly the one you get by doing a single integral. Why are there two y's? Well, it's because the height is given to be $y$ and the width of each rectangle is also $y$ (Essentially making the cross section a square). There, you'll get the volume of one slice of the solid to be $V=y \cdot y \cdot dx$. You could just take cross sections perpendicular to x instead of y (LaTeX ahead)
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