Can one consider Archimedes as an inventor of the integral...

Key Concepts

Method of Exhaustion

The method of exhaustion is an early technique used by mathematicians, including Archimedes, to determine areas and volumes by inscribing and circumscribing sequences of shapes. This approach approximated the desired quantity to arbitrary precision and is considered a precursor to the formal development of integration in calculus.

Archimedes

Archimedes was an ancient Greek mathematician and engineer whose work laid important foundations for many areas of mathematics. His contributions include conceptualizing methods close to what we now call calculus, such as calculating areas and volumes using limiting processes, even though he did not develop the formal framework of calculus as understood today.

Integral Calculus

Integral calculus is a branch of mathematics that focuses on the accumulation of quantities, the concept of the area under curves, and the computation of antiderivatives. It formalizes the process of integration, which computes total values from rates of change, and is one of the two principal branches of calculus alongside differential calculus.

Transcript

00:01 In this question we are going to focus on explaining the archimedes principle in our own words and we will also see one example where archimedes principle can be applied.

00:10 So here we can say that according to archimedes principle, the buoyant force on any object is equal to weight of fluid displaced by that very object.

00:22 So it will be here weight of and then it will be here fluid and now we can say displaced by the object.

00:31 Now what is the buoyant force? we can here discuss the meaning of buoyant force with the help of a figure.

00:38 So suppose we are having a ball which is being submerged inside water.

00:43 So this is our container and inside it we are having water filled up to a certain height and here we are having a ball.

00:50 Now on this very ball there will be two main forces.

00:54 One will be in the downward direction that is the weight.

00:58 And one in the upward direction that will be basically buoyant force.

01:02 We call it thrust, right? so it can be here shown as such.

01:08 Now we are here saying that buoyant force on any object which is being submerged inside a liquid is equal to weight of fluid displaced by the object.

01:19 Now we can here discuss that how this very principle can be derived.

01:24 Suppose we are having density to be equals to mass divided with volume.

01:29 We know that, right? that density is equal to mass divided with volume.

01:33 Hence we can say that mass will be equals to density that is row multiplied with volume.

01:40 Hence, the value of weight can be here written to be equals to mass times acceleration, right? so it will be here a -w -c -e -l -e -r -a -t -i -o -n.

01:54 And here we will be having acceleration due to gravity.

01:57 So w will be equals to mass times g...

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