Seed Question: A spring has the following |FSP| vs graph. We take the absolute value of x FSP so we can ignore signs. Can you find the magnitude of the work this spring would do in being stretched from x0 = 0 to xf = x? Exploration: Using the graph above: Introduction to Integral Calculus: Make a small rectangle near the center of the graph of small width (∆x) whose height represents the average force in that interval. How tall should that rectangle be? The area of this slender rectangle represents the tiny work done Find in terms ∆W. ∆W of k, x and (∆x). If we made a whole series of rectangles like this and added them up, we could find the approximate total area, and the approximate total work! This is all very impressive, but kind of clunky. Here comes the amazing part: shrink the ∆x’s to infinitesimal dx’s, then add up all the infinitesimal rectangular areas to find the exact total area and work. Rewrite as in terms of and (dx). dw k, x ∆W Here is how you integrate a polynomial from 0 to x: Now integrate dW to find an algebraic solution to the total work done by the spring. Did you get the same answer you got to the seed question?
Seed Question: A spring has the following |FSP| vs x graph. We take the absolute value of FSP so we can ignore signs. Can you find the magnitude of the work this spring would do in being stretched from x0 = 0 to xf = x? Exploration: Using the graph above: Introduction to Integral Calculus: Make a small rectangle near the center of the graph of small width (∆x) whose height represents the average force in that interval. How tall should that rectangle be? The area of this slender rectangle represents the tiny work done ∆W. Find ∆W in terms of k, x and (∆x). If we made a whole series of rectangles like this and added them up, we could find the approximate total area, and the approximate total work! This is all very impressive, but kind of clunky. Here comes the amazing part: shrink the ∆x’s to infinitesimal dx’s, then add up all the infinitesimal rectangular areas to find the exact total area and work. Rewrite ∆W as dw in terms of k, x and (dx). Here is how you integrate a polynomial from 0 to x: Now integrate dW to find an algebraic solution to the total work done by the spring. Did you get the same answer you got to the seed question?