For new 2008 add maths syllabus students
This question combines the modulus function with integration.
Perhaps you will like to solve it.
Question
Integrate l x^2 + 2x -3 l dx from x = 0 to x = 2
[The sign in blue is the modulus sign]
Answer
Step 1 : Sketch the curve y = x^2 + 2x -3
The curve will be an "U-shaped" curve.
The curve will cut the x-axis at x = 1 and x = - 3
Part of the curve will be below the x-axis.
Step 2 : Sketch the curve y = l x^2 + 2x - 3 l
Part of the curve that is previously below the x-axis will be reflected to be above the x-axis.
It will be observed that the area from x = 1 to x = 2 is positive.
The area from x = 0 to x = 1 will be positive too but it will be difficult to integrate to find this area using the modulus function, so instead we will integrate the curve y = x^2 + 2x - 3 that is below the x-axis from x = 0 to x = 1. These two areas are the same as these areas are the reflection of one another.
Step 3 : Find the area
Integrate l x^2 + 2x -3 l dx from x = 0 to x = 2
= Integrate x^2 + 2x -3 dx from x = 1 to x = 2
+ Integrate [- ( x^2 + 2x -3)] dx from x = 0 to x = 1
= 4
So, you have learnt the trick of solving this type of question. So, now you know how to solve it if this type of question comes out in the "O" level add maths exam.
Integrate x^2 + 2x -3 dx from x = 1 to x = 2
Don't we always write the bigger integer at the top? For example, we go from F(2) - F(1) if not the area would be negative.
Dear Darrick_3658,
When we use the integral symbol, we put the x = 2 on top and x = 1 below.
However, when we describe in it words, we say integrate x^2 + 2x -3 dx from x = 1
to x = 2. (Pan Pacific Add Maths Textbook Page 427).
To summarise,
F(b) - F(a) = Integral symbol (x= b to x = a) f(x) dx = area of the curve y = f(x)
from x = a to x = b
Thanks.
Regards.
ahm97sic