1. 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

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

3. 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

4. WBHS Advanced Programme Mathematics
Integration (Riemann Sums)
A few very useful and important sigma expansions …..
Think … (n times) = n …. 
Think … (n times) = ….
Think … (n times) = ….
Think … (n times) = ….

5. 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.

6. 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

7. 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

8. 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

9. 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

10. 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!!

11. 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.

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