Quick questions on Numerical methods: change of sign, iteration, Newton-Raphson and the trapezium rule - Edexcel A-Level Maths

An equation rearranged into the form $x = g(x)$ gives an iteration $x_{n+1} = g(x_n)$. Starting from a sensible estimate, repeated application converges to a root when the iteration is suitable, often shown as a staircase or cobweb diagram. The same equation can usually be rearranged into $x = g(x)$ in several ways, and only some of them converge: convergence happens when the gradient of $g$ near the root has magnitude less than $1$. A staircase diagram appears when the iterates approach the root from one side, and a cobweb diagram when they alternate either side as they close in.

The rule approximates the area under the curve by a row of trapezia, so it is exact only when the curve is a straight line. For a curve that bends upward (concave up) the trapezia lie above the curve, so the rule overestimates the area; for a curve that bends downward it underestimates. Using more strips makes each trapezium hug the curve more closely and reduces the error.

Show that $f(x) = x^3 - x - 1$ has a root between $1$ and $2$. [2 marks]

Use the trapezium rule with two strips to estimate $\int_0^2 x^2\,dx$. [3 marks]