Calculus  Integration  Approximation methods.
Trapezoidal Rule  Test Yourself 1.
1. 

2.  Note there are two triangles with which we will use the normal "area of a triangle" formula. The green triangle is however below the xaxis with a negative y value for its height. Calculating the area therefore gives a negative value. As Area cannot be negative, we take the absolute value before adding: 

3.  
4. (i)  (ii) (iv) Area = 1×1.25 + 1×3.25 = 4.5 u^{2}. (v) Comparing the areas between the curve and the respective areas in the rectangles shows the estimate using the rectangles is very close to the actual area under the curve. By observation, the area under the rectangles is slightly less than the total area under the curve. 

3 function values  5. (i) y = 2x  x^{2} with the area bounded by the curve and the xaxis


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More than 3 function values.  10. (i)
(ii) 

11. Find an approximation for .
 
12. (i)
(ii) Volume = 23,500m^{2} × 200 m = 4,700,000 m^{3} = 4.7 million cubic metres. 

13. (i) (ii)
As the sin 2x curve goes beneath the x axis and we need to calculate the area NOT evaluate the integral, we need to do the calculation in two parts  from π/3 to π/2 and then from π/2 to π. Then reverse the sign of the 2nd calculation before adding. 

14. Use 4 subintervals to find an approximation to .


Diagram  15.
(i)
Distance = 0.5×(15.8 + 33 + 38) = 634.9 m. (ii) The calculated distance would be slightly less than the actual distance travelled. The straight lines joining successive dots to make the trapezia would all be under the actual curve although reasonable close to it. 

16.
Adding across the 2nd row to get sums of successive pairs: Area = 2.5×(8 + 20 + 22 + 10) = 150 m^{2}. 

17. (i)
(ii) Volume = 74.75 × 2 m × 60 min = 8,970 m^{3}/hour = 8,970 Ml/hour. 

18.
