Question

What is the relationship between area and volume? Is it similar to calculating an area by slicing it into narrower and narrower lengths?

Answer 1

The area, measured in square unit, represents  the extent of space taken up by a plane object or curve. Volume, measured in cubic unit s, refers to the amount of space inside the three-dimensional object enclosed by a closed surface, i.e. it determines the space that the shape contains.

Recall what we do while calculating definite integral. Actually, definite integral gives area under the curve say, y= f(x). We slice that area into strips of length y( i.e. height) and in order to make this slice as rectangular, we take its width very small( i.e. like dx). Thus, calculating area of one strip ( that would be y×dx) and integrating it over the limits of integral will give resultant area.

( I am not very sure what you want to ask, but I am providing something which is very usual in calculus). There is a concept in calculus called " rotation of curve about given axis", this is somewhat similar to like finding area under the curve by integrating. Here, we revolve given curve about desired line to get a solid which would result into volume of that solid. If we revolve a circle given by x​​​​​​2​​​ + y​​​​​​2 = a​​2 ​​​​​​, about either x axis or y axis, we will get a sphere whose volume will be (4π/3)a​​​​​​3 .

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