Dynamic Calculus
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The definite integral
The applet below shows a section of the curve \(y = f(x)\). The interval \([a, b]\) is divided into \(n\) equal sub-intervals, each of width \(\delta x\). What is the limit of the sum of the rectangles as \(\delta x\) approaches zero?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
Consider the area \(A\) bounded by the curve, the \(x\)-axis and the ordinates \(x = a\) and \(x = b\). If \(h\) = sum of the areas of the lower rectangles and \(H\) = sum of the areas of the upper rectangles, then \(h\delta x < A < H\delta x\).
- Increase the number of sub-intervals n by moving the slider to the right and confirm the following:
- the width of each rectangle \(\delta x\) becomes very small
- the sum of the areas of the rectangles becomes very close to \(A\).
- What relationship exists between the values in each row of the table in the top right corner as \(n\) increases? Confirm this relationship for other positions of the ordinates \(a\) and \(b\) by clicking and dragging these points.
- Using your answer to question (2), write down an expression for the sum of the areas of the rectangles, when \(n\) is large.
- As n approaches ∞, \(\delta x\) approaches zero and the sum of the areas of the rectangles approaches \(A\). (In other words, the area \(A\) is equal to the limit of the sum of the areas of the rectangles as \(\delta x\) approaches zero.) Express this statement using the appropriate summation notation. Check your answer by clicking the 'show notes' checkbox.
[The last line of the notes introduces a special symbol to represent the operation of finding the area under a curve, \[A = \int_a^b {f(x)dx} \]
It is read as 'the definite integral of \(f(x)\) as \(x\) goes from the lower limit \(x = a\) to the upper limit \(x = b\)']