WBHS Advanced Programme Mathematics

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Presentation transcript:

WBHS Advanced Programme Mathematics Integration Process of finding the area between the x-axis and a curve, on an interval between two x-values Common technique is to use rectangles (Riemann Sums) Trapeziums also work very well conceptually

WBHS Advanced Programme Mathematics Integration (Riemann Sums) a b x1 x2 xi = a + i(b – a)/n n rectangles Rectangle width = (b – a)/n

Integration (Riemann Sums) WBHS Advanced Programme Mathematics Integration (Riemann Sums) b x1 x2 xi = a Area of n rectangles = sum of (rectangle width × rectangle heights) = (rectangle width) × (sum of rectangle heights) = × n rectangles Rectangle width = (b – a)/n

WBHS Advanced Programme Mathematics Integration (Riemann Sums) A few very useful and important sigma expansions ….. Think 1 + 1 + 1 + … (n times) = n ….  Think 1 + 2 + 3 + … (n times) = …. Think 1 + 4 + 9 + … (n times) = …. Think 1 + 8 + 27 + … (n times) = ….

WBHS Advanced Programme Mathematics Integration (Riemann Sums) Calculate the Riemann Sum for from x = −2 to x = 4. Use 18 rectangles of equal width. 1. 2. 3. −2 4 xi 4.

WBHS Advanced Programme Mathematics Integration (Riemann Sums) Calculate the Riemann Sum for from x = −2 to x = 4. Use 18 rectangles of equal width. 4. −2 4 xi

WBHS Advanced Programme Mathematics Integration (Riemann Sums) Calculate the area below the curve and above the x-axis, on the interval from x = −2 to x = 4. −2 4 xi

WBHS Advanced Programme Mathematics Integration (Riemann Sums) Calculate the area below the curve and above the x-axis, on the interval from x = −2 to x = 4. −2 4 xi

WBHS Advanced Programme Mathematics Integration (Riemann Sums) Calculate the area above the curve and below the x-axis, on the interval from x = −2 to x = 4. xi −2 4

WBHS Advanced Programme Mathematics Integration (Riemann Sums) Calculate the area above the curve and below the x-axis, on the interval from x = −2 to x = 4. xi −2 4 Clearly this needs some interpretation!!

WBHS Advanced Programme Mathematics Integration (Riemann Sums) Calculate the area above the curve and below the x-axis, on the interval from x = −2 to x = 4. Interpretation: “Negative areas” are simply those that lie below the x-axis. We don’t try to prevent the algebra in the Riemann Sum from giving us a “negative area”, we simply accommodate this in our understanding. For this reason, we must calculate areas that lie above and below the x-axis separately from one another. Areas that lie below the x-axis will result in negative Riemann Sums but we just ‘ignore” the negative sign for these. We then simply add a number of positive areas together to get a total area enclosed by a curve and the x-axis.

WBHS Advanced Programme Mathematics Integration (Riemann Sums) Important notation for the total area enclosed between a curve and the x-axis: The “definite integral”