![]() The following depicts a side view of the triangular slice. Thus, the length of the base of an arbitrary cross sectional triangular slice is: So for that arbitrary #x#-value we have the associated #y#-coordinates #y_1, y_2# as marked on the image: In order to find the volume of the solid we seek the volume of a generic cross sectional triangular "slice" and integrate over the entire base (the circle) The grey shaded area represents a top view of the right angled triangle cross section. The antiderivative of that and then evaluate a definite integral.Consider a vertical view of the base of the object. Have a polynomial in terms of y and we know how to take If you multiply both of these terms by y, well then you're just goin'a And that's all they asked us to do to express the volumeĪs a definite integral, but this is actually a definite integral that you could solve without a calculator. So integrate from y is equal to zero to y is equal to 12. Cross-sections perpendiculai to the x-axis are isosceles right triangles with hypotenuse in the base. The base of a solid S is an elliptical region with boundary curve 16x2 +9y2 144. V 256,3 256 3 x Find the volume V of the described solid S. Just goin'a integrate from y equals zero to y is equal to 12. Parallel cross-sections perpendicular to the base are squares. Volume of the whole figure, it's gonna look something like, something like that, we're ![]() Of infinitesimal depth, d y depth, is going to be y times nine minus y squared over 16 d y. The volume of this little slice right over here So negative y squared over 16 is equal to x minus nine. And then, let's see, we could multiply both sides by negative one. Y squared over 16 isĮqual to nine minus x. So you get y over four is equal to the square root of nine minus x. So here we just have to solve for x, so one way to do this is, let's see, we can square both sides of, oh, actually let's divideīoth sides by four. And so what you do isĮxpress x in terms of y. Now if wanna integrate with respect to y, we want everything in terms of y. The volume of this little slice is going to be y times x times d y. 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!! and the cross sections taken are perpendicular to the x-axis are isosceles right triangles with bases lying along the base of the circle. 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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