Simpson's rule estimation

1. Use Simpson's rule with four strips to estimate the value of the integral

$$\int _ { 0 } ^ { 3 } \mathrm { e } ^ { \cos x } \mathrm {~d} x$$

4 The integral I is defined by $$I = \int _ { 0 } ^ { 13 } ( 2 x + 1 ) ^ { \frac { 1 } { 3 } } d x$$

1. Use integration to find the exact value of I .
2. Use Simpson's rule with two strips to find an approximate value for I. Give your answer correct to 3 significant figures.

3

1. Find, in simplified form, the exact value of \(\int _ { 10 } ^ { 20 } \frac { 60 } { x } \mathrm {~d} x\).
2. Use Simpson's rule with two strips to find an approximation to \(\int _ { 10 } ^ { 20 } \frac { 60 } { x } \mathrm {~d} x\).
3. Use your answers to parts (i) and (ii) to show that \(\ln 2 \approx \frac { 25 } { 36 }\).

7

1. Find the exact value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).
2. Use Simpson's rule with two strips to show that an approximate value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\) can be expressed in the form \(m + n \sqrt [ 3 ] { 36 }\), where the values of the constants \(m\) and \(n\) are to be stated.
3. Use the results from parts (i) and (ii) to find an approximate value of \(\sqrt [ 3 ] { 36 }\), giving your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers. \section*{Question 8 begins on page 4.}

1 Use Simpson's rule, with five ordinates (four strips), to calculate an estimate for $$\int _ { 0 } ^ { \pi } x ^ { \frac { 1 } { 2 } } \sin x d x$$ Give your answer to four significant figures.
[0pt] [4 marks]

2 Use Simpson's rule with 5 ordinates ( 4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.