A Level Maths: Integration - Exam Question
Introduction
n this article we are going to explore a question on Integration which is part of A Level maths at year 12 and year 13. It is a past exam question and it is worth 9 marks, so quite some working and thinking is needed.
Quite often at maths A Level you are being tested on your ability to apply your knowledge to a variety of questions. The concepts stay the same, but there are occasions where you need to be working against what you may be used to.
This is why the best way to get good at maths is to do as many questions as possible.
The Question
The Solution
You are given the gradient function and the only thing that you can do here is to integrate in order to obtain an expression for y remembering that it is a indefinite integral and so you need to remember to include the constant c.
You given the value of x and the value of y which the curve passes through i.e. (1,8) and so substituting these values in will give the following:
\begin{aligned} 8 & =1-\frac{a}{2}+3+c \\ 8 & =4-\frac{a}{2}+c \rightarrow c=4+\frac{a}{2} \\ \therefore y & =x^2-\frac{a}{2 x^2}+3 x+\left(4+\frac{a}{2}\right) \end{aligned}The question tells what the area is between two x values and so you are essentially performing the following integral:
\int_1^3 y d x=30This can now be done as follows:
\begin{aligned} & \int_1^3 x^2-\frac{a}{2} x^{-2}+3 x+\left(4+\frac{a}{2}\right) d x=30 \\ & {\left[\frac{x^3}{3}+\frac{a}{2 x}+\frac{3}{2} x^2+\left(4+\frac{a}{2}\right) x\right]_1^3=30} \\ & {\left[\frac{3^3}{3}+\frac{a}{6}+\frac{3}{2}(3)^2+\left(4+\frac{a}{2}\right)(3)\right]-\left[\frac{1}{3}+\frac{a}{2}+\frac{3}{2}+4+\frac{a}{2}\right]=30} \\ & \left(9+\frac{a}{6}+\frac{27}{2}+12+\frac{3}{2} a\right)-\left(\frac{1}{3}+\frac{a}{2}+\frac{3}{2}+4+\frac{a}{2}\right)=30 \\ & \frac{69}{2}+\frac{5}{3} a-\frac{35}{6}-a=30 \rightarrow \frac{86}{3}+\frac{2}{3} a=30 \\ & \frac{2}{3} a=\frac{4}{3} \\ & \therefore a=2 \end{aligned}You will see why this question is worth a total of 9 marks. There is quite a lot of work to do but the question shows you that you will not always be integrating an expression with known values. Here we have an expression with an unknown value, the area is known, so it shows how the need to be able to apply your mathematical knowledge is the key to success at A Level Maths.
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