SL 5.11—Definite integrals, areas under curve onto x-axis and areas between curves

The antiderivative of $3x^2-4x+1$ is $F(x)=x^3-2x^2+x$.

${\left[x^3-2x^2+x\right]}_0^2=F(2)-F(0)$

$F(2)=(2{)}^3-2(2{)}^2+2=8-8+2=2$

$F(0)=0-0+0=0$

${\int }_0^2(3x^2-4x+1)dx=2-0=2$

We need to integrate $3x^2-4x+1$ from $0$ to $2$ using the power rule.

If someone simply evaluates $f(2)=12-8+1=5$ or makes arithmetic errors summing term integrals incorrectly, they might get $6$. This is not correct.

A student might forget to subtract $F(0)$ correctly or mishandle the $-4x$ term's antiderivative, arriving at $4$. Checking: $F(2)=8-8+2=2\ne 4$.

Getting $0$ would suggest the integral is zero, which would require equal positive and negative areas. Since $f(0)=1>0$ and $f(2)=5>0$, the function is not symmetric about zero on $[0,2]$, so this is incorrect.

The correct evaluation gives $F(2)-F(0)=2-0=2$.

$x^2-9=0\Rightarrow x=\pm 3$. On $(-3,3)$, $f(x)<0$; on $(3,4)$, $f(x)>0$.

The definite integral ${\int }_{-3}^4(x^2-9)dx$ computes the net signed area: the negative region from $-3$ to $3$ partially cancels the positive region from $3$ to $4$.

$\text{Total Area}=\left|{\int }_{-3}^3(x^2-9)dx\right|+\left|{\int }_3^4(x^2-9)dx\right|$

The definite integral gives a net area smaller in magnitude than the total area, confirming the second option is correct.

The question asks about the conceptual difference between the definite integral value and the total geometric area when a curve crosses the $x$-axis.

$f(0)=-9<0$, so the function is clearly not always non-negative on $[-3,4]$. This option is false.

To convert net area to total area, you take $|{\int }_{-3}^3|+|{\int }_3^4|$, not 'plus twice'. This formula is incorrect.

Being a polynomial has no bearing on whether the definite integral equals total area. Any function that dips below the $x$-axis will have net area $\ne$ total area.

The definite integral gives net signed area, which is less than the total area when the curve is below the $x$-axis on part of the interval — confirming the second option.