Calculus  Integration  Approximation methods.
Trapezoidal Rule  Test Yourself 1.
Geometric methods.  1. Sketch the line y = 3x.
(ii) Find the area between the line y = 3x and the xaxis from x = 0 to x = 4 using a direct geometric method. Answer.Area = 24 u^{2}. 

2. (i) Sketch the line y = 5  2x for the domain [1, 5].
(ii) Find the area between the line y = 5  2x and the x axis between 

3. Find the exact area between the semicircle and the coordinate axes in the first quadrant.
Answer.Exact area = 4π. 

4. (i) Draw the parabola y = x^{2} + 1.
(ii) Draw a rectangle with its base on the xaxis and its top horizontal side going through the parabola at x = 0.5 and extending from x = 0 to x = 1. (iii) Draw a second rectangle with its base on the xaxis and its top horizontal side going through the parabola at x = 1.5 and extending from x = 1 to x = 2. (iv) Using each of these two rectangles, determine the approximate area between the parabola and the x axis between x = 0 and x = 2. (v) Is the area you have calculated an overestimate or an under estimate of the actual area? Explain your answer.


3 function values.  5. (i) Sketch the graph of y = 2x  x^{2}.


6. Use the Trapezoidal rule with 3 function values to determine an approximation of the area between the curve f(x) = 2^{x} and the xaxis between x = 0 and x = 4. Answer.Approximately 25 u^{2}. 

7. Use the Trapezoidal rule with 3 function values to determine an approximation of the area between the curve f(x) = log_{e}x and the xaxis between x = e and x = 3e. Answer.Approximately 8.814. 

8. Find the approximate area under the function y = 1 + 2 log_{10} x between the values x = 1 and x = 7 using the Trapezoidal rule with 3 function values.
Answer.Approximately 12.147. 

9. Use the Trapezoidal Rule with three function values to find an approximation to correct to 2 significant figures. Answer.Approximately 0.98. 

More than 3 function values.  10.(i) Using the x values given in the following table, complete the values for f(x) = x^{2} log_{e} x to three decimal places.
(ii) Use the Trapezoidal Rule with the values in the table to obtain an approximation to . Answer.Area = 21.719 (approx). 

11. Use the Trapezoidal Rule with 5 function values as shown in the table below to find an approximation for .


12. A team of geologists is measuring the depth to which they must drill to take a core sample containing lithium oxide. They drill along a northsouth line every 100 metres. The depths to which they drill each hole are recorded in the table below.
Answer. Area = 1.133 u^{2} (ii) Volume = 4.7 million cubic metres. 

13. (i) Sketch the curve y = sin 2x between x = 0 and x = 2π.
(ii) Using the trapezoidal rule, find the area between the curve 

14. Given the function f(x) = 3^{cos x}, apply the Trapezoidal Rule with 4 subintervals to find an approximation to .
Answer.Approx 4.52. 

Diagram  15.
The diagram shows the graph of a particle's velocity v m/sec at time t seconds. Answer.Distance = 634.9 m (approx). 

16. The diagram below shows a paddock ADB bounded by a river AB and three parallel fences.
The distances of each fence from the end of the paddock to the river are: GH = 8 m; CD = 12 m; EF = 10 m. All distances AG, GC, CE and EB are 5 m. Use the Trapezoidal Rule to find the approximate area of the paddock. Answer. Area = 150 m^{2}. 

17. At a certain location along a river, the width is 20 metres. Measurements of the depth across the river at this point have been taken and these measurements are recorded on the crosssectional diagram below.
Answer.(i) Area = 74.75 m^{2} (ii) 8,970 m^{3} in 1 hour or 8,970 ML/hpur. 

18. The following diagram shows the graphs of y = ln(x + 1) and .
Complete the following table (using your calcuator) and then calculate the area of the shaded region between the two curves from x = 1 to x = 3.
