CHAPTER 72 AREAS UNDER AND BETWEEN CURVES

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CHAPTER 7 AREAS UNDER AND BETWEEN CURVES EXERCISE 8 Page 77. Show by integration that the area of the triangle formed by the line y, the ordinates and and the -ais is 6 square units.

1 CHAPTER 7 AREAS UNDER AND BETWEEN CURVES EXERCISE 8 Page 77. Show by integration that the area of the triangle formed by the line y, the ordinates and and the -ais is 6 square units. A sketch of y is shown below. yd d [ ] 6 6 square units. Sketch the curve y + between and. Determine by integration the area enclosed by the curve, the -ais and ordinates and. Use an approimate method to find the area and compare your result with that obtained by integration. A sketch of y + is shown below, John Bird

y d ( + ) d [ + ] (7 ) ( ) + square units Width of interval.5 8 X.5.5..5..5. y +..75..75. 7.75. 9.75 8 Hence, using Simpson s rule, (.5). 8 (.75.75 7.75 9.75) (...) + + + + + + + + Area ( ) (.5) 6.

2 y d ( + ) d [ + ] (7 ) ( ) + square units Width of interval.5 8 X y Hence, using Simpson s rule, (.5). 8 ( ) (...) Area ( ) (.5) [ ] If a greater number of intervals is chosen the area would be close to square units. Find the area enclosed between the curve, the horizontal ais and the given ordinates: y 5;, A graph of y 5 is shown below. 5 yd 5d () (.5) 7.5 square units. Find the area enclosed between the curve, the horizontal ais and the given ordinates: y + ;, A sketch of y + is shown below, John Bird

3 ( ) 6 yd + d square units 5. Find the area enclosed between the curve, the horizontal ais and the given ordinates: y sin θ; θ, θ π A sketch of y sin θ is shown below. / / / y d sin θ d [ cos θ] cos ( cos ) π π π π square unit 6. Find the area enclosed between the curve, the horizontal ais and the given ordinates: θ t + e t ; t, t t ( t+ e t )dt + et + e ( + e ) 8.89 square units 5, John Bird

4 7. Find the area enclosed between the curve, the horizontal ais and the given ordinates: y 5 cos t; t, t 6 π A sketch of y 5 cos t is shown below. π /6 /6 /6 d π 5cos d sin π sin π sin sin π 6 y t t [ t] square unit 8. Find the area enclosed between the curve, the horizontal ais and the given ordinates: y ( )( );, A sketch of y ( )( ) is shown below Shaded area ( + ) ( + ) ( ( ) d ( )( ) d d d ( ) ( ).67 square units 6, John Bird

5 EXERCISE 8 Page 77. Find the area enclosed between the curve, the horizontal ais and the given ordinates: y ;, A sketch of y is shown below. yd yd d d [(8 ) 8)] 6 square units. Find the area enclosed between the curve, the horizontal ais and the given ordinates: y ;, A sketch of y, i.e. y is shown below. 7, John Bird

6 π /6 yd d [ ln ] ln ln ln 5.55 square units. The force F newtons acting on a body at a distance metres from a fied point, is given by: F +. If work done Fd, determine the work done when the body moves from the position where m to that when m. Work done ( ) 7 Fd + d N m. Find the area between the curve y and the -ais. y ( ). When y, and. A sketch of y is shown below 6 ( ) d ( ).67 square units 5. Determine the area enclosed by the curve y 5 +, the -ais and the ordinates and. Find also the area enclosed by the curve and the y-ais between the same limits. A sketch of y 5 + is shown below 8, John Bird

7 5 yd 5 + d + (5 + 6) () ( ) 5 square units The area enclosed by the curve y 5 + (i.e. y ), the y-ais and the ordinates y and 5 y 7 (i.e. area ABC in the sketch above) is given by: Area y 7 7 y 7 ( y ) d y d y ( y ) d y y square units 6. Calculate the area enclosed between y 5 and the -ais. y 5 ( ) 5 ( 5( + ) Hence, when y, or 5 or When, y ( )() 6 A sketch of the graph y 5 is shown below ( ) ( ) d 5 d ( ) + ( ) (.9666) ( ) 7.8 square units 9, John Bird

8 7. The velocity v of a vehicle t seconds after a certain instant is given by: v (t + ) m/s. Determine how far it moves in the interval from t s to t 5 s Distance moved area under v/t graph v t ( t ) t [ t t] d + d + (5 + ) ( + ) m 8. A gas epands according to the law pv constant. When the volume is m the pressure is 5 kpa. Find the work done as the gas epands from m to a volume of m given that work done v pd v. v pv k When v m and p 5 kpa then k pv 5 kpa m kn 5 k m m 5 knm 5 kj k v 5 v v Work done pd v d v d v [ 5ln v] ( 5ln 5ln) v 5 ln 69. kj, John Bird

9 EXERCISE 85 Page 775. Determine the coordinates of the points of intersection and the area enclosed between the parabolas y and y. y and y i.e. y or y 9 Equating y values gives: 9 i.e. 7 i.e. 7 i.e. ( 7) Hence, or 7 from which, 7 and 7 Thus, and When, y and when, y Hence, (, ) and (, ) are the points of intersection of the two curves A sketch of the curves is shown below. d d 9 / / + 9 ( ) square units, John Bird

10 . Sketch the curves y + and y 7 and determine the area enclosed by them. The two curves intersect when + 7 i.e. + i.e. ( + )( ) i.e. when and The two curves are shown below. Area enclosed by curves ( ) ( ) ( ) (7 )d + d (7 ) + d d square units or.8 square units. Determine the area enclosed by the curves y sin and y cos and the y-ais. A sketch of y sin and y cos is shown below, John Bird

11 When sin cos then sin cos, i.e. tan and tan 5 or π π π π π rad / / ( cos sin ) d [ sin + cos ] sin + cos ( sin + cos ) ( ) ( + ). square units. Determine the area enclosed by the three straight lines y, y and y + 5 y + 5 and y intersect when + 5 i.e. when 5 5 i.e. when y + 5 and y intersect when + 5 i.e. when 5.5 i.e. when The three straight lines are shown below d + ( + 5) d 5 () ( ), John Bird

12 sq units ( ), John Bird

CHAPTER 71 NUMERICAL INTEGRATION

CHAPTER 71 NUMERICAL INTEGRATION CHAPTER 7 NUMERICAL INTEGRATION EXERCISE 8 Page 759. Evaluate using the trapezoidal rule, giving the answers correct to decimal places: + d (use 8 intervals) + = 8 d, width of interval =.5.5.5.75.5.65.75.875.