How do I find the area under a wave/quadratic?

Question

The roots A and B are (-1,0) and (2,0) respectively.

Answer 1

You have correctly done the first part. To find the area of the shaded region, you need to have been taught how to at least integrate polynomial functions using the fundamental theorem of calculus.

In particular here, since the graph is that of the function defined by $2x^3-9x^2+3x+14$ on the real axis, you want to integrate this function between the points A and B, namely from $x=-1$ to $x=2.$ But why would this integral give you the area? Recall that for a nonnegative continuous function defined on some interval of the form $[a,b],$ where $a,b$ are real numbers, you can interpret the integral of this function on this interval as being the area bounded by the graph of this function, the $x$-axis, and the lines $x=a,\,x=b.$ You see this by dividing this region using ordinates over the interval $[a,b]$ and approximating the thin slices defined by this partition by rectangles. We can estimate the area of each of these by multiplying by an appropriate value of the function and the length of the interval over which the slice stands. This is what is called a Riemann sum. When we take finer and finer partitions, the sum of these rectangles converges to the area of the region, which as we already know is also the integral of the function over this interval.

So, again, what you need do is integrate your cubic over the interval $[-1,2].$ To do this, recall that if $x^n$ is a polynomial, then one of its primitives is $x^{n+1}/(n+1).$