Trapezium rule estimation

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 $$\int _ { 1 } ^ { 5 } \left| 2 ^ { x } - 8 \right| \mathrm { d } x$$

\includegraphics{figure_4} The diagram shows the curve with equation \(y = \sqrt{1 + e^{0.5x}}\). The shaded region is bounded by the curve and the straight lines \(x = 0\), \(x = 6\) and \(y = 0\).

1. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures. [3]
2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [4]

\includegraphics{figure_12} The figure above shows the curve \(C\) with equation $$f(x) = \frac{x+4}{\sqrt{x}}, \quad x > 0.$$

1. Determine the coordinates of the minimum point of \(C\), labelled as \(M\). [5]

The point \(N\) lies on the \(x\) axis so that \(MN\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(MN\) and the straight line with equation \(x = 1\).

1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [3]