Slippery areas

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Take a look at this diagram.

The area of the blue region is \(53\) and the area of the orange region is \(125\).

Use this to explain why \[\int_{15}^{30}3f(2x-45)\,dx=67.\]

Now try to evaluate each of the following integrals.

1. \(\int_{-4}^{-2}\sqrt{x+5}\,dx\)

2. \(\int_{0.2}^{1.3}e^{2x}\,dx\)

Give some examples of other functions that you can integrate in this way.