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

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

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

Since this matches the y-coordinate of point N, we conclude that point N lies on the curve C.

To compute the integral, we integrate term by term:

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

where $C$ is the constant of integration.

The area of the region R can be found by evaluating the definite integral from L to M where the curve is above the x-axis:

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

Using the integral from part (c), we calculate:

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

Calculating this gives:

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

After evaluating the two parts, we can compute the final area of region R.