4. (a) Find the values of (i) $\int _ { - 1 } ^ { 1 } \left( x ^ { 2 } - x \right) \mathrm { d } x$, (ii) $\int _ { - 1 } ^ { 1 } \left( x ^ { 3 } + x ^ { 2 } - 2 x \right) \mathrm { d } x$. (b) Sketch the graph of $y = x ^ { 2 } - x$ and indicate which difference in areas is represented by your answer to (a)(i). (c) Find the total area (measured positively) that lies between the graphs of $y = x ^ { 2 } - x$ and $y = x ^ { 3 } + x ^ { 2 } - 2 x$ between $x = - 1$ and $x = 1$. (d) The answers to (a)(i) and (a)(ii) are related in a particular way. Explain how the relationship can be seen without working out any integrals.
4. (a) Find the values of\\
(i) $\int _ { - 1 } ^ { 1 } \left( x ^ { 2 } - x \right) \mathrm { d } x$,\\
(ii) $\int _ { - 1 } ^ { 1 } \left( x ^ { 3 } + x ^ { 2 } - 2 x \right) \mathrm { d } x$.\\
(b) Sketch the graph of $y = x ^ { 2 } - x$ and indicate which difference in areas is represented by your answer to (a)(i).\\
(c) Find the total area (measured positively) that lies between the graphs of $y = x ^ { 2 } - x$ and $y = x ^ { 3 } + x ^ { 2 } - 2 x$ between $x = - 1$ and $x = 1$.\\
(d) The answers to (a)(i) and (a)(ii) are related in a particular way. Explain how the relationship can be seen without working out any integrals.\\