Find the area of the shaded
Let’s use the given figure to first
determine the equations of each of the lines that bound this region. First, we have the curve 𝑦 equals
three 𝑥 squared plus four 𝑥 minus two. We also have the vertical lines 𝑥
equals one and 𝑥 equals two. But rather than our region being
bounded by the 𝑥- or 𝑦-axis, the final line which bounds this region is the line
𝑦 equals five.
Now if we were just looking to find
the area bounded by the quadratic graph, the lines 𝑥 equals one and 𝑥 equals two
on the 𝑥-axis, we could evaluate the definite integral from one to two of three 𝑥
squared plus four 𝑥 minus two with respect to 𝑥. Notice though that this would
include the area of the rectangle below the line 𝑦 equals five. And so to find the area of just the
pink region, we’d need to subtract the area of this rectangle from our integral.
The area of this rectangle isn’t
tricky to find. It has a vertical height of five
units and a width of one unit. That’s two minus one. So its area is just five multiplied
by one, or five. This then gives us one method that
we can use. We evaluate the definite integral
to find the area all the way down to the 𝑥-axis and then subtract the area of the
orange rectangle. Working through the integral would
give an answer of six square units.
But there is actually another way
that we could answer this question. The area below the line 𝑦 equals
five can also be found using integration. It’s equal to the integral from one
to two of five with respect to 𝑥. And using the linear property of
integration, we could then express this as a single integral. If you were to form this integral,
it would give the same result.