C2 2016 2 Edexcel

(a)

We have at x = 1:

y = 8 - 2^{1-1} = 8 - 2^0 = 8 - 1 = 7

(b)

The step length is 1, so using the trapezium rule:

\begin{align*}\int_0^4 \left(8 - 2^{x - 1}\right) \mathrm dx & \approx \frac 1 2 \left(7.5 + 0 + 2(7 + 6 + 4)\right) \\ & = 20.75\end{align*}

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Note that:

\displaystyle \mathrm {area}(R) = \int_0^4 \left(8 - 2^{x - 1}\right) \mathrm dx - \mathrm{area}(\Delta ABO)

The area of ABO is:

\displaystyle \frac 1 2 \times |OA| \times |OB| = \frac 1 2 \times 4 \times 7.5 = 15

(noting that A = (4,0) and B = (7.5,0))

So:

\displaystyle \mathrm {area}(R) \approx 20.75 - 15 = 5.75

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