how to integrate $ \int_\frac74^2 \lvert 1-x\rvert dx $ manully?

Question

I calculated it on https://www.integral-calculator.com/ $$ \int_{7/4}^{2}\left\lvert 1 - x\right\rvert\mathrm{d}x $$ and got the answer but I could not get the answer right when substituting. Also, it would be very helpful if there were another way to do the integration instead of the solution given by the site

Answer 1

The best way to get a grasp of the situation (in my opinion) would be to graph the integrand over the given interval:

The vertical lines represent the bounds of your integration, and the red graph is $f(x) = |1-x|$. Recalling that the integral just gives the signed area under a curve, if you can find the area of the green trapezoid, you can find the value of the integral.

(Of course, you can also just note as Adam did that, on $(7/4,2)$, $|1-x| = x-1$, and so you can perform a more standard integral that way. I prefer this geometric explanation, but the algebraic idea is also nice.)

Answer 2

$\lvert 1-x\rvert = -1+x$ when $x\geq1$