Problem Description : Find area of the shaded region ABCD in the figure below.
Solution :
Observation : We begin with the area of the shaded region as shown in the above figure which can be obtained easily using integration as
Now the required Ar(ABCD) = AR(OBCO) - AR(DOAD) - AR(BAOB) + AR(AOA) where each of the areas of regions OBCO, DOAD, BAOB, AOA can be obtained by reusing the above observation accordingly.
Let us now see a similar illustration of the reusability of a result
Problem Description : Find the area of ∆leABC in the below figure
Observation : Consider the ∆le PQR as shown in the figure below
Therefore,
Now, from the observation,
the required Ar(∆leABC) =AR(∆leAFE) - AR(∆leBFD) +AR(∆leCED)
the required Ar(∆leABC) =AR(∆leAFE) - AR(∆leBFD) +AR(∆leCED)
I remember solving this problem when I was preparing for IIT-JEE. Obviously, I solved it using the naive approach which took over two pages to arrive at the result. However, to our awe, Prof. G.N. Subramaniam (who posed this problem in class) showed us an alternative approach: working in transformed coordinates. Define u := (x^2/y) and v := (y^2/x). This transformation converts the integral domain to a simple rectangle and the result is got in few steps. However, one needs to evaluate the Jacobian matrix of the trasformation that relates the two integrals.
ReplyDeleteIn those days we (those who managed to understood this approach) were amazed by these transformed spaces and that you get the result in a few steps! Now in the hindsight, I think Koundinya's approach is simpler, more elegant/clever (effective usage of existing results) and appropriate (for 11/12 class students).
Dear Dr.Prshanth..
ReplyDeleteits greate to here from u that u remember even the classroom disussions of u r 12th std ..
surprising..
prof G N S may wanted to motivate in his class at that time certain students of tranfomations of spaces.. but thats a classical example to tranformations to obtain results in particular to evaluate certain integrals