In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 12 - 2022 - Paper 2

The area A can be expressed as:

[ A = \int_{2}^{4} \frac{(x - 2)(x - 4)}{4 \sqrt{x}} , dx ]

First, let's simplify the integrand:

[ \frac{(x - 2)(x - 4)}{4 \sqrt{x}} = \frac{x^2 - 6x + 8}{4 \sqrt{x}} ]

Next, we can express ( \sqrt{x} ) in terms of its exponent:

[ \sqrt{x} = x^{1/2} ]

Thus,

[ A = \int_{2}^{4} \frac{x^2 - 6x + 8}{4 x^{1/2}} , dx = \int_{2}^{4} \left( \frac{1}{4} x^{3/2} - \frac{3}{2} x^{1/2} + 2 \right) , dx ]

Now, we can integrate term by term:

1. ( \int \frac{1}{4} x^{3/2} , dx = \frac{1}{4} \cdot \frac{2}{5} x^{5/2} = \frac{1}{10} x^{5/2} )
2. ( \int -\frac{3}{2} x^{1/2} , dx = -\frac{3}{2} \cdot \frac{2}{3} x^{3/2} = -x^{3/2} )
3. ( \int 2 , dx = 2x )

Thus, the definite integral becomes:

[ A = \left[ \frac{1}{10} x^{5/2} - x^{3/2} + 2x \right]_{2}^{4} ]

Calculating the definite integral:

[ A = \left( \frac{1}{10} (4)^{5/2} - (4)^{3/2} + 2(4) \right) - \left( \frac{1}{10} (2)^{5/2} - (2)^{3/2} + 2(2) \right) ] Which simplifies to:

[ A = \left( \frac{1}{10} \cdot 32 - 8 + 8 \right) - \left( \frac{1}{10} \cdot 4 - 2 + 4 \right) ]

Calculating each term gives:

1. At x = 4: ( \frac{32}{10} - 8 + 8 = \frac{32}{10} = 3.2 )
2. At x = 2: ( \frac{4}{10} - 2 + 4 = \frac{4}{10} + 2 = 2.4 )

Combining gives:

[ A = 3.2 - 2.4 = 0.8 ]

Finally, express the area in the required form:

Since ( 0.8 = \frac{4}{5} = \sqrt{2} ) and there are no additional constants,

Thus, we conclude that:

[ A = \sqrt{2} (with \ a = 2, \ b = 0) ]