Calc - Integ - Approx

Geometric methods.

1. Sketch the line y = 3x.

(ii) Find the area between the line y = 3x and the x-axis from x = 0 to x = 4 using a direct geometric method.

Answer.Area = 24 u².

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
x = 1 and x = 5 using direct geometric methods.

Answer.Area = 2.25 + 6.25 = 8.5 u².

3. Find the exact area between the semi-circle

and the coordinate axes in the first quadrant.
Answer.Exact area = 4π.

4. (i) Draw the parabola y = x² + 1.

(ii) Draw a rectangle with its base on the x-axis 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 x-axis 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 - x².

(ii) Shade the area bounded by the curve and the x-axis where y ≥ 0.

(iii) Use the Trapezoidal rule with 3 function values to find the area you have shaded.

(iv) Is your answer an over-estimate or an under-estimate of the exact value of the area under the curve?

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
x-axis between x = 0 and x = 4.
Answer.Approximately 25 u².

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

8. Find the approximate area under the function y = 1 + 2 log₁₀ 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) = x² log_e x to three decimal places.

x	2	2.5	3	3.5	4
f(x)

(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

x	1	1.5	2	2.5	3
f(x)	11.2	17.8	9.3	4.1	11.6

Answer.Integral = 21.3.

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 north-south line every 100 metres. The depths to which they drill each hole are recorded in the table below.

Spacing	0	100	200	300	400
Depth	40	80	50	70	30

(i) Calculate the area of the depth to which they drill along this line.

(ii) If the site is 200 metres wide, find the approximate volume of soil they have to remove to expose the lithium deposits.

Answer. Area = 1.133 u²
(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
y = sin 2x and the x-axis between x = π/3 and π using 5 function values (answer correct to 3 decimal places).

Answer. Area = 1.133 u^{2^.}

14. Given the function f(x) = 3^{cos x}, apply the Trapezoidal Rule with 4 sub-intervals 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.

(i) Use the trapezoidal rule with 4 function values to approximate the distance the particle travelled in the first 3 seconds.

(ii) Is the estimate you found for the distance the particle travelled more of less than the exact answer? Give a reason for your answer.

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 m².

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 cross-sectional diagram below.

(i) Use the trapezoidal rule to estimate the cross-sectional area at this point.

(ii)The river is flowing at a rate of 2 m/min. Calculate the total volume of water (in m³ and to the nearest megalitre) flowing past this point in the river in one hour.

Answer.(i) Area = 74.75 m²
(ii) 8,970 m³ 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.
(Answer correct to 2 decimal places).

	1	2	3
y = ln(x + 1)

Difference

Answer. Area = 2.972 u².