Limit of sum as integral

4. a. Express \(\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x \quad\) as an integral.
b. Hence show that $$\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x = \ln k$$ where \(k\) is a constant to be found.

5. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \sqrt { x }\) The point \(P ( x , y )\) lies on the curve.
The rectangle, shown shaded on Figure 3, has height \(y\) and width \(\delta x\).
Calculate $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 9 } \sqrt { x } \delta x$$

1 Fig. 1 shows some spreadsheet output concerning the values of a function, \(\mathrm { f } ( x )\). \begin{table}[h] \end{table} The formula in cell B2 is ==EXP(-(A2\^{}2)) and equivalent formulae are in cells B3 to B6. The formula in cell C 2 is \(= \mathrm { B } 2\).
The formula in cell C3 is \(\quad = \mathrm { C } 2 + \mathrm { B } 3\). Equivalent formulae are in cells C4 to C6.

1. Use sigma notation to express the formula in cell C5 in standard mathematical notation.
2. Explain why the same value is displayed in cells C 5 and C 6. Now suppose that the value in cell C2 is chopped to 3 decimal places and used to approximate the value in cell C2.
3. Calculate the relative error when this approximation is used. Suppose that the values in cells B4, B5 and B6 are chopped to 3 decimal places and used as approximations to the original values in cells B4, B5 and B6 respectively.
4. Explain why the relative errors in these approximations are all the same.

14

1. Express \(\lim _ { \delta y \rightarrow 0 } \sum _ { 0 } ^ { h } \left( h ^ { 2 } - y ^ { 2 } \right) \delta y\) as an integral.
2. Hence show that \(V = \frac { 2 } { 3 } \pi r ^ { 2 } h\), as given in line 41 .