June 2022 Paper 2 Q14

OCR MEICurrent spec8 marksIntegrationNumerical Methods

14 Fig. 14.1 shows the curve with equation \(y = \dfrac{1}{1+x^2}\), together with 5 rectangles of equal width.

Fig. 14.1: the curve y = 1/(1+x^2) for x from 0 to 1 through points A, B, C, D, E, F at x = 0, 0.2, 0.4, 0.6, 0.8, 1, with 5 rectangles of width 0.2 lying under the curve and the horizontal lines at the left-hand heights
Fig. 14.1

Fig. 14.2 shows the coordinates of the points A, B, C, D, E and F.

PointABCDEF
\(x\)00.20.40.60.81
\(y\)10.961540.862070.735290.609760.5

Fig. 14.2

(a) Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to show that a lower bound for \(\displaystyle\int_0^1 \frac{1}{1+x^2}\,\mathrm{d}x\) is 0.7337, correct to 4 decimal places. [2]
(b) Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to calculate an upper bound for \(\displaystyle\int_0^1 \frac{1}{1+x^2}\,\mathrm{d}x\) correct to 4 decimal places. [2]
(c) Hence find the length of the interval in which your answers to parts (a) and (b) indicate the value of \(\displaystyle\int_0^1 \frac{1}{1+x^2}\,\mathrm{d}x\) lies. [1]

Amit uses \(n\) rectangles, each of width \(\dfrac{1}{n}\), to calculate upper and lower bounds for \(\displaystyle\int_0^1 \frac{1}{1+x^2}\,\mathrm{d}x\), using different values of \(n\). His results are shown in Fig. 14.3.

\(n\)102040
upper bound0.809980.797790.79162
lower bound0.759980.772790.77912

Fig. 14.3

(d) Find the length of the smallest interval in which Amit now knows \(\displaystyle\int_0^1 \frac{1}{1+x^2}\,\mathrm{d}x\) lies. [2]
(e) Without doing any calculation, explain how Amit could find a smaller interval which contains the value of \(\displaystyle\int_0^1 \frac{1}{1+x^2}\,\mathrm{d}x\). [1]
‹ Previous74 of 86 in IntegrationNext ›