Building blocks

## Solution

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 $\int_{15}^{30}3f(2x-45)\,dx=67.$

Combining these ideas explains why $\int_{15}^{30}3f(2x-45)\,dx=67.$

By considering relationships between $3f(2x-45)$, $10-3f(x)$ and $f(x)$, we were able to piece together enough information to find the complicated-looking integral $\int_{15}^{30}3f(2x-45)\,dx.$

More generally, if we can relate a complicated integral to a simpler one, we may be able to use this to evaluate the more complicated one.

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.

In the GeoGebra files below, the blue graphs show the functions we want to integrate and the orange graphs show a related function that we already know how to integrate. (The corresponding regions are shaded in blue and orange respectively.) We’ve also added a red function, which is controlled by the slider. Move the slider to see how the blue graph can be transformed into the orange graph, and back again.