We first need to draw the pictures and identify the intersections of the curves. We can do this by using the calculator. We will insert the functions \(x^3-4x^2+1\) and \(x-3\) in the memory locations
y1
and
y2
. Choose \(\blacklozenge\) and then
F1
. In the field
y1
type the text
x^3-4x^2+1
. Type the text
x-3
in the field
y2
. The graphs are obtained by pressing
F2
and selecting the zoom level
6: ZoomStd
.
Our next task is to identify the intersection points of the two graphs. This can be obtained by returning to the home screen with the button
HOME
. By typing
solve(y1(x)=y2(x),x)
in the input line we obtain that the \(x\) coordinates of the intersection points are \(-1\), \(1\), and \(4\).
By analyzing the graph we see that the function \(y_1(x)\) is above the function \(y_2(x)\) on the interval \([-1,1]\). The function \(y_2(x)\) is below \(y_1(x)\) on the interval \((1,4)\). Therefore, the area between the graphs is
\begin{eqnarray*}A&=&\int_{-1}^1\left(y_1(x)-y_2(x)\right)\,dx+\int_1^4\left(y_2(x)-y_1(x)\right)\,dx.
\end{eqnarray*}
We will use the calculator to evaluate the integrals and their sum.
We will evaluate the sum of the integrals by first typing
F3
and then choosing the integral from the menu. The characters \(\int(\) will appear in the input line. We complete the input line by writing
y1(x)-y2(x),x,-1,1)
. Then we press the button
+
. The next step is to write another integral (
F3
and then choose integral from the menu). The input line will receive the characters \(\int(\). We need to add
y2(x)-y1(x),x,1,4)
. If we press the button
ENTER
, then we receive the answer \(\frac{253}{12}\). If we press the button \(\blacklozenge\) before the button
ENTER
, then the output becomes \(21.0833\). Therefore, the correct answer is A.