You should already be aware that the definite integral of
is equivalent to the signed area under the curve of
![\displaystyle\int_0^1f(t)\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_0%5E1f%28t%29%5C%2C%5Cmathrm%20dt)
corresponds the area of the triangle on the left, and
![\displaystyle\int_1^4f(t)\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_1%5E4f%28t%29%5C%2C%5Cmathrm%20dt)
is the area of the trapezoid in the middle. You should know how to compute the area of these shapes.
We end up with
![\displaystyle\int_0^4f(t)\,\mathrm dt=-\frac{1\times2}2+\frac{(1+3)\times2}2=3](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_0%5E4f%28t%29%5C%2C%5Cmathrm%20dt%3D-%5Cfrac%7B1%5Ctimes2%7D2%2B%5Cfrac%7B%281%2B3%29%5Ctimes2%7D2%3D3)
since the triangle has height 2 and base 1, and the trapezoid has height 2 with two base lengths 1 and 3.
Edit: The area under the curve is highlighted red in the attachment below. It's the area between the curve and the horizontal axis, but it's also signed, meaning the area <em>below</em> the axis is negative, and the area <em>above</em> the axis is positive.