Simpson's rule estimation

8 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-4_787_742_276_719} The diagram shows part of the curve \(y = \ln \left( 5 - x ^ { 2 } \right)\) which meets the \(x\)-axis at the point \(P\) with coordinates \(( 2,0 )\). The tangent to the curve at \(P\) meets the \(y\)-axis at the point \(Q\). The region \(A\) is bounded by the curve and the lines \(x = 0\) and \(y = 0\). The region \(B\) is bounded by the curve and the lines \(P Q\) and \(x = 0\).

1. Find the equation of the tangent to the curve at \(P\).
2. Use Simpson's Rule with four strips to find an approximation to the area of the region \(A\), giving your answer correct to 3 significant figures.
3. Deduce an approximation to the area of the region \(B\).

8 \includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-4_538_1443_262_351} The diagram shows the curve with equation \(y = x ^ { 8 } \mathrm { e } ^ { - x ^ { 2 } }\). The curve has maximum points at \(P\) and \(Q\). The shaded region \(A\) is bounded by the curve, the line \(y = 0\) and the line through \(Q\) parallel to the \(y\)-axis. The shaded region \(B\) is bounded by the curve and the line \(P Q\).

1. Show by differentiation that the \(x\)-coordinate of \(Q\) is 2 .
2. Use Simpson's rule with 4 strips to find an approximation to the area of region \(A\). Give your answer correct to 3 decimal places.
3. Deduce an approximation to the area of region \(B\).

8 The definite integral \(I\) is defined by $$I = \int _ { 0 } ^ { 6 } 2 ^ { x } \mathrm {~d} x$$

1. Use Simpson's rule with 6 strips to find an approximate value of \(I\).
2. By first writing \(2 ^ { x }\) in the form \(\mathrm { e } ^ { k x }\), where the constant \(k\) is to be determined, find the exact value of \(I\).
3. Use the answers to parts (i) and (ii) to deduce that \(\ln 2 \approx \frac { 9 } { 13 }\).

1. (a) Use Simpson's rule with 4 intervals to find an estimate for

$$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \sin ^ { 2 } x } \mathrm {~d} x$$ Give your answer to 3 significant figures. Given that \(\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \mathrm { sin } ^ { 2 } x } \mathrm {~d} x = 3.855\) to 4 significant figures,
(b) comment on the accuracy of your answer to part (a).

The integral \(I\) is defined by $$I = \int_0^{13} (2x + 1)^{\frac{3}{2}} \, dx.$$

1. Use integration to find the exact value of \(I\). [4]
2. Use Simpson's rule with two strips to find an approximate value for \(I\). Give your answer correct to 3 significant figures. [3]