Area involving absolute values

2 Use the trapezium rule with three intervals to find an approximation to $$\int _ { 0 } ^ { 3 } \left| 3 ^ { x } - 10 \right| \mathrm { d } x$$

1 Use the trapezium rule with 3 intervals to estimate the value of $$\int _ { 0 } ^ { 3 } \left| 2 ^ { x } - 4 \right| \mathrm { d } x$$

1 Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 5 } \left| 2 ^ { x } - 8 \right| \mathrm { d } x$$

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 5 )$$ Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between \(x = 0\) and \(x = 2\) and is bounded by \(C\), the \(x\)-axis and the line \(x = 2\).
(9)

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = x ( x + 4 ) ( x - 2 )$$ The curve \(C\) crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).

1. Write down the \(x\)-coordinates of the points \(A\) and \(B\). The finite region, shown shaded in Figure 3, is bounded by the curve \(C\) and the \(x\)-axis.
2. Use integration to find the total area of the finite region shown shaded in Figure 3.

7 \includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-3_579_557_858_794} The diagram shows part of the curve \(y = x ^ { 2 } - 3 x\) and the line \(x = 5\).

1. Explain why \(\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 3 x \right) \mathrm { d } x\) does not give the total area of the regions shaded in the diagram.
2. Use integration to find the exact total area of the shaded regions.

10 In this question you must show detailed reasoning.
The equation of a curve is \(y = 12 x ^ { 3 } - 24 x ^ { 2 } - 60 x + 72\).
Determine the magnitude of the total area bounded by the curve and the \(x\)-axis.

5 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = x ^ { 3 } - 4 x\). \includegraphics[max width=\textwidth, alt={}, center]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-05_499_695_404_251} Determine the total area enclosed by the curve and the \(x\)-axis.

9

1. 1. Find $$\int \left( 4 x - x ^ { 3 } \right) d x$$ 9
2. (ii) Evaluate $$\int _ { - 2 } ^ { 2 } \left( 4 x - x ^ { 3 } \right) \mathrm { d } x$$

   9 Using a sketch, explain why the integral in part (a)(ii) does not give the area enclosed

   between the curve \(y = 4 x - x ^ { 3 }\) and the \(x\)-axis.

   between the curve \(y = 4 x - x ^ { 3 }\) and the \(x\)-axis. [2 marks] 9
3. Find the area enclosed between the curve \(y = 4 x - x ^ { 3 }\) and the \(x\)-axis.