3 The diagram shows a sketch of part of the curve with equation y=9-3x-5x^2-x^3 and the line with equation y=4-4x The line cuts the curve at the points A(-1,8) and B(1,0) Find the area of the shaded region between AB and the curve.

## Step 1:Firstly, we can calculate the area by integrating the height from the x-coordinate of point A to the x-coordinate of point B. We calculate the height by subtracting the y-value of the line at x from the y-value of the curve at x.Formally, we have a height , where is the y-value of the curve at x and = y-value of the line at x. From the given problem we know \(f(x) = 9 - 3x - 5x^{2} -x^{3}\) and \(g(x) = 4 - 4x\) ## Step 2:Now by simplicity and commutative rules and due to \(H(x)\) being a differene, we can rearrange to get \(H(x) = (-x^{3} - 5x^{2} + x + 5)\)## Step 3: Finally, we use an integral to calculate the area as follows for x values within the limits of A and B which are -1 and 1 respectively.### ## Step 4: Now, integral of \(H(X)\) within the limits of -1 and 1 gets us to;### ## Step 5: Proceeding with above gets as the result for the integral to be### ## Step 6:Further simplification turns into:###