Complete the table with the values of y corresponding to x = 2 and 2.5 - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 4

Using the trapezium rule, the formula for the area is:

$A = \frac{1}{2} \times (b_1 + b_n) + \sum_{i=1}^{n-1} b_i$

Where b are the y-values. Here b_1 = 16.5, b_n = 0, and the other y-values are: 7.361, 4, 2.31, 1.278, 0.556.

Calculating:

$A = \frac{1}{2}(16.5 + 0) + (7.361 + 4 + 2.31 + 1.278 + 0.556)$ $A = 8.25 + 31.505 = 39.755$

Thus, the approximate area is 11.88 (to 2 decimal places).

To find the exact area of R, we integrate the function:

$A = \int_{1}^{4} \left( \frac{16}{x^2} - \frac{x}{2} + 1 \right) \ dx$

Calculating the integral:

$= \left[ -\frac{16}{x} - \frac{x^2}{4} + x \right]_{1}^{4}$

Evaluating the bounds:

$= \left[ -4 - 4 + 4 \right] - \left[ -16 - \frac{1}{4} + 1 \right]$ $= [ -4 ] - [ -16.25 ] = 12.25$

Thus, the exact area is 12.25.