The curve C has equation $y = x^2 - 5x + 4$ - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 4

To show that the point N(5, 4) lies on curve C, we substitute $x = 5$ into the equation:

$y = 5^2 - 5(5) + 4 = 25 - 25 + 4 = 4$

Since $y = 4$ when $x = 5$, the point N(5, 4) indeed lies on the curve.

We can find the integral:

$\\int (x^2 - 5x + 4) \,dx = \frac{x^3}{3} - \frac{5x^2}{2} + 4x + C$

where C is the constant of integration.

To find the area of region R, we need to calculate the definite integral of the curve between points L and M. The limits are from $x = 1$ to $x = 4$:

$\text{Area} = \int_1^4 (x^2 - 5x + 4) \,dx\,$

Substituting the limits using the indefinite integral found in part (c):

$= \left[\frac{x^3}{3} - \frac{5x^2}{2} + 4x\right]_1^4$

Calculating this gives:

$= \left(\frac{4^3}{3} - \frac{5(4^2)}{2} + 4(4)\right) - \left(\frac{1^3}{3} - \frac{5(1^2)}{2} + 4(1)\right)$

Evaluating both parts results in the exact value of the area of R.