![]() We do realize that we need to write the in terms of because we are integrating with respect to. Yay! Now we are down to integration, our food. If we take infinitely many cylinders, we get the integral, Let’s sum up all these n cylinders (like we summed up all the rectangles when we were introducing the integral) to get the total volume. The volume of an arbitrary cylinder with thickness (or height), is given byīut the radius is actually the -value at that arbitrary value, so the volume is We then sum up all the volumes of the cylinders across the entire cone, the thinner the cylinders the better. We start by chopping off a typical circular cylinder at some arbitrary value. We will put the cone with base radius and length(height) in the Cartesian plane like below: Summer is coming, we would like to know how much ice cream those cones can hold! Let’s go right ahead and chop the cone into infinitely many cylinders. How do we calculate the volume of a non-cylindrical solid figure (how do we calculate the area of a non-rectangular object?)? We chop the solid figure into infinitely many cylinders (we chop the object into infinitely many rectangles). Unfortunately, not all the solid figures that we come across everyday are cylinders. This is because they have two identical flat ends and the same cross-section from one end to the other. You should be able to calculate the volumes of the cylinders below (yes, they are all cylinders.)Ĭylinders are nice, we only need to multiply the cross-sectional area by the height/length to find the volume. ![]() I suggest that we start by looking at the solids whose volume we know very well. We have seen how integration can be used to solve the area problem, in this post we are going to see how we can use a similar idea to solve the volume problem. I would like us to discuss one of the important applications of integration. Hi there again, I have not written a post in while, here goes my second post.
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