AUC20 Calculate the area between the curve y = x2 + x - 2 and the x-axis between x = 0 and x = 2.
Solution Do not write down the integral of x2 + x - 2 with the prescribed limits and perform the integration to find the answer.
If you do, you will get the problem wrong!
|
Sketch the graph of this function. Factor y = x2 + x - 2 = (x - 1)(x + 2) and notice that the curve crosses the x-axis at x = 1 and x = -2. Look at the limits of the integration. At x = 0, y = -2. At x = 2, y = 4. With this information the curve can be sketched. More detail for the sketch is not necessary. |
|
The area between this curve and the x-axis has to be calculated in two pieces corresponding to the two areas marked A1 and A2.
Notice the integrand is written as [0-(x2 + x - 2)]. This statement is the "top curve," y = 0, minus the "bottom curve" y = x2 + x - 2. Writing the integrand this way, top curve minus bottom curve, keeps the area positive. This is the preferred way of writing the problem. It will prove very helpful in more complicated problems.
Now find the second area, A2. The integrand x2 + x - 2 would be viewed as top curve minus bottom curve, which is 0.
The total area between the curve and the x-axis is the sum of these two areas.
Standard Mistake: Don't make this mistake. If you take the integral of x2 + x - 2 between the limits of 0 and 2 you will get an answer that is equal to A2 - A1. It will look great but it is wrong. Take the integral of x2 + x - 2, use the limits 0 and 2 and verify that this is the difference in the areas and the incorrect answer. This is the kind of problem that math profs use to separate the A's from the B's.
We've had A's and we've had B's. A's are better.
