Tips and commentary for the first problem submitted through our website chat!
This is a more challenging question, towards the end of an AS level paper on the new 2017 syllabus. It is "synoptic" in that you have to draw upon skills from several parts of the course: Algebra | Calculus | Curve Sketching
The 2 areas must be considered separately, and will be obtained by integrating the function. However the equation given cannot be integrated in its current form and you will need to expand the 3 brackets to obtain a cubic polynomial. The most efficient way to do this here would be to multiply the first 2 brackets then multiply by the third bracket.
Then the integration can be performed. When we integrate between limits (for x) it's called definite integration, but here the limits are not given. So how can we find the limits?..
They are the points where the curve intersects the x-axis (called roots), and here they can be written down straight from the factorised form as x=-2, x=2, x=-1.
So the area enclosed above the x-axis will be obtained between limits of -2 and -1, whereas the area below the x-axis uses limits -1 and 2. Remember that an area below the x-axis will drop out of the maths as a negative value, so ignore the negative sign when combining to obtain total area.
Note that if you wish to check your workings here using the integration function on a scientific calculator, you must consider each area separately.
Comments