Dynamic Calculus
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Area between two curves
The applet below shows the straight line \(y = mx\) and the cubic function \(y = {x^3}\). How do we calculate the area bounded by the two curves?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
Investigate the function of the 'slider' by moving it left and right.
- Calculate the area bounded by the straight line \(y = g(x)\), the \(x\)-axis and the ordinates \(x = 0\) and \(x = b\). Confirm your answer by clicking the 'Area under g(x) = mx' checkbox.
- Calculate the area bounded by the cubic function \(y = f(x)\), the \(x\)-axis and the ordinates \(x = 0\) and \(x = b\). Confirm your answer by clicking the 'Area under f(x) = x³' checkbox.
- Using your answers to questions (1) and (2), how would you calculate the area between the two curves in the first quadrant? Calculate this area then check your answer by clicking the relevant checkbox.
- What type of functions are \(y = g(x)\) and \(y = f(x)\)? What relationship exists between the area between the two curves in the first quadrant and the area between the two curves in the third quadrant?
- Calculate the total area bounded by the two curves then check your answer by clicking the relevant checkbox.
- If it is not obvious which curve lies above the other over an interval, how do we ensure that the answer returned by integration is a positive area?