ABSTRACT
When y = 0, x = 0 or (x + 2) = 0 or (x − 4) = 0, i.e. when y = 0, x = 0 or −2 or 4, which means that the curve crosses the x-axis at 0, −2, and 4. Since the curve is a continuous function, only one other co-ordinate value needs to be calculated before a sketch of the curve can be produced. When x = 1, y = −9, showing that the part of the curve between x = 0 and x = 4 is negative. A sketch of y = x3 − 2x2 − 8x is shown in Fig. 38.2. (Anothermethod of sketchingFig. 38.2 would have been to draw up a table of values.)
Shaded area
(x3 − 2x2 − 8x)dx − ∫ 4
0 (x3 − 2x2 − 8x)dx
= [ x4
4 − 2x
3 − 8x
− [ x4
4 − 2x
3 − 8x
= ( 6 2 3
) − ( −422
) =4913 square units
2. the area between the curves y = x + 1 and y = 7− x .
