Applications 11.1 Areas under curves A11.1 Areas under curves Before ou start You should be able to: calculate the value of given the value of in algebraic equations of curves calculate the area of a trapezium. bjectives You will be able to calculate an estimate for the area of the region bounded b a curve, the -ais and two lines parallel to the ais. You will be able to interpret the area under a velocit-time graph as a distance travelled. Wh do this? Working out areas under curves is ver important when studing the thermodnamics of engines. Get Read 1 In each case calculate the value of when i ii a b c 6 Calculate the area of the trapezium. 7 Ke Points An estimate of the area of the region between a curve and the -ais can be calculated b splitting the region into trapeziums and finding the total area of the trapeziums. The area of the region between a velocit-time (v/t) curve and the time (t)-ais is equal to the distance travelled. v t The area of this region is equal to the distance travelled. Eample 1 Here is a curve. Calculate an estimate of the area bounded b the curve, the -ais, and the lines and. 1
Chapter 11 Areas under curves A B C D Then add all these areas together. A: B: C: D: ( 8) (8 1) 11 (1 ) 19 ( ) 9 Split the area of the region into trapeziums and work out the area of each trapezium. A quicker wa to calculate the sum of the areas of the trapezium is to factorise the sum of the epressions: ( 8 8 1 1 ) ( 8 1 ) [ (8 1 ) ] 6 square units Total area 6 square units Eample Here is a sketch of the graph of for values of from 0 to. 1 Calculate an estimate for the area of the region shown shaded in the diagram. 0 1 0 8 18 An estimate for the area is 8 square units. Split the region into trapeziums each of width 1 unit. Work out the lengths of the parallel sides of each trapezium b using the equation of the curve. (The first trapezium will be a triangle.) Eample A particle moves in one direction in a straight line. Its velocit v m/s after t seconds is given b v t t. Calculate an estimate of the distance it has travelled during the first seconds. t 0 1 0 8 1 The distance travelled area between the curve and the time ais, from t 0 to t. Using the trapezium rule, an estimate for this area is 8 square units. So the distance travelled is 8 metres.
A Applications 11.1 Areas under curves Eercise 11A 1 Find the area of the regions bounded b the curve, the -ais and the lines shown parallel to the -ais. a b c 1 0 1 1 1 1 1 1 B d e 1 f 0 0 0 1 1 0 1 1 Draw the curve with equation for values of from to. Calculate an estimate for the area between the -ais, the curve and the lines and. a Cop and complete the table of values for the curve with equation 8. 0 1 6 7 1 b Draw the graph. c Calculate an estimate for the area between the curve, the -ais and the lines 1 and 6. The graph gives information about the velocit of a particle during the first seconds of its motion. a Calculate an estimate for the distance travelled during the first seconds. b State, with a reason, whether our answer to a is an overestimate or underestimate of the true distance travelled. Velocit m/s 1 1 1 1 0 t
Chapter 11 Areas under curves A The graph shows the velocit, v m/s, with which a racing car has travelled during t seconds. v 0 90 70 t a Calculate an estimate of the distance the racing car travelled during these seconds. b Calculate an estimate of the average speed of the car for these seconds. 6 The velocit, v m/s, of a particle at time t seconds is given b v t t. a Calculate an estimate of the distance it has travelled after seconds. b Calculate an estimate of the average speed of the particle during the first seconds. Review The approimate area under a graph can be found b approimating the region b a set of trapeziums and finding their total area. The area under a velocit-time graph is equal to the distance travelled.
Answers Answers Chapter 11 A11.1 Get Read answers 1 a i ii 18 b i ii 1. c i ii 6 16. Eercise 11A 1 a 1 b 6 c 79 d 9 e 170 f 900 a metres b overestimate as the sloping edges of the trapeziums are alwas above the curve. a metres b 9.7 m/s 6 v 1 6 t a Distance travelled is approimatel metres. b Average speed is approimatel 16 m/s. Area is approimatel 6 square units 0 1 6 0 7 1 1 16 1 1 b 1 1 6 c 6 square units