Aims: By the end of this chapter, you
will be able to use definite integral to find
(i) area underneath a curve, and
(ii) area bounded by two curves.
The shaded area = ∫_{a}^{b} y(x) dx. |
If the shaded area of a curve y(x) is separated into two parts
at x = c such that one part is above the x-axis and the other is
below that x-axis then the total shaded area is In the example 3 above, the value of c is where the curve y(x) crosses the x-axis. The value c is π/2 in this particular example. |
GDC Besides using the function fnInt( function to find
definite integral. We can also use graphical method. In this case,
enter our equation into [Y= ].
We then plot the graph using [GRAPH].
To calculate a definite integral as in example 1 above, we press [2nd][TRACE]
for CALC and then press
7: ∫ f(x) dx. |
Key: 0.463 (3.s.f.) |
Key: (b) (0.750,2.44) & (3.25,2.44); (c) 7.48 (3 s.f.) |
displacement (s)→ [ds/dt] → velocity (v) → [dv/dt] → acceleration (a) displacement (s ) ← [∫ v dt] ← velocity (v) ← [∫ a dt] ← acceleration (a) |
It is a good idea to have a sketch of the function v.
Here is a screen capture from the Ti calculator. (a) Displacement over the first 6 seconds = ∫ ^{6} [3 - 5sin(t/2)] dt. ≈ -1.90 (3 s.f.) |