= [ + + + 4 + + + + 3 + + + + 0] = square OR 8,59 square units Activity 4 1. Using rectangles, calculate the areas described in each of the following cases. Use a partition with a sensible number of rectangles. Make sure each rectangle has the same width. Make a rough sketch for each case. a) = 2x + 7 between the x-axis, and = 2 and = 3. f(x) x x – b) = x2 bounded by the x and = 4 and = 1. k(x) x x –axis – – c) bounded by the and = 1 and = 4. j(x) = , with x > 0, x-axis x x d) = + 2, between the and = 5 and = 3. Is this area an or m(x) x2 x-axis x x – overestimate of the true area? e) = between the x-axis and the two x-intercepts. Can you work out the exact area p(x) under this curve? How accurate was your partitioning method? f) = 2 sinx, between the x-axis and = and = . h(x) x x 2. Create an Excel spreadsheet to approximate the area bounded by the curve = + 4 on the interval [–1 ; 2] for 3 different rectangle widths. t(x) –x2 3. Determine the area under the function = , for [– 1 ; 6], by dividing the given g(x) x interval into five trapeziums. Let the height of each trapezium be 1 unit. Use the following formula to assist you to calculate the area of each trapezium: Area = (sum of parallel sides) height 3.1 How does this method compare with method of dividing the given interval into rectangles? What areas that fall below the x-axis? Consider the following example: = 2)(x h(x) x(x + 2) –
= [ + + + 4 + + + + 3 + + + + 0] = square OR 8,59 square units Activity 4 1. Using rectangles, calculate the areas described in each of the following cases. Use a partition with a sensible number of rectangles. Make sure each rectangle has the same width. Make a rough sketch for each case. a) f(x) = 2x + 7 between the x-axis, and x = – 2 and x = 3. b) k(x) = x2 bounded by the x –axis and x = – 4 and x = – 1. c) j(x) = , with x > 0, bounded by the x-axis and x = 1 and x = 4. d) m(x) = x2 + 2, between the x-axis and x = – 5 and x = 3. Is this area an or overestimate of the true area? e) p(x) = between the x-axis and the two x-intercepts. Can you work out the exact area under this curve? How accurate was your partitioning method? f) h(x) = 2 sinx, between the x-axis and x = and x = . 2. Create an Excel spreadsheet to approximate the area bounded by the curve t(x) = –x2 + 4 on the interval [–1 ; 2] for 3 different rectangle widths. 3. Determine the area under the function g(x) = , for x [– 1 ; 6], by dividing the given interval into five trapeziums. Let the height of each trapezium be 1 unit. Use the following formula to assist you to calculate the area of each trapezium: Area = (sum of parallel sides) height 3.1 How does this method compare with method of dividing the given interval into rectangles? What areas that fall below the x-axis? Consider the following example: h(x) = x(x – 2)(x + 2)
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