Find the area beneath the curve with equation f(x) = 3x^2 - 2x + 2 when a = 0 and b = 2
This question is an example of integrating to find the area underneath a curve between two points. We begin by intergrating the equation. Firstly, to integrate 3x2 we increase the indice/power by one unit and then divide whatever the x was multiplied with by this new power term. With this example, the x2 would increase to x3 and then we would divide the 3 by this new power term (3/3 = 1) so we would be left with 1x3 or simply x3. We would do the same with the -2x which would leave us with -x2. For the constant, 2, we would multiply this by x, giving us 2x. We can check our answer by differentiating it (the opposite of integration); if it leaves us with the same equation as that given in the question, then our answer is correct. For integration questions within a limit, we do not include the + c (+ constant) in our answer.
The second part of our solution involves us calculating the area between the curve and between the limits of 0 and 2. Our answer from the previous part of the solution can be written like so: [x3 - x2 + 2x]02. We then substitute the limits into the xs and subtract the bottom limit subtituted equation from the top: (23 - 22 +22) - (03 - 02 + 20) = 8 - 0 = 8.
The area beneath the curve between the two limits is 8.
