Find the area of the region under the graph of $f$ on $[ a , b ]$.
$$f ( x ) = x ^ { 2 } - 2 x + 4 ; \quad [ - 1,2 ]$$
Select the correct answer.
A) 3
B) $- 12$
C) 12
D) $- 3$

Question

Quizplus · Accepted Answer

To find the area under the graph of a function on a given interval, we need to integrate the function over that interval. Thus, we need to evaluate:

$$\int_{-1}^2 (x^2 - 2x + 4) dx$$

Using the power rule of integration, we get:

\begin{align*}
\int_{-1}^2 (x^2 - 2x + 4) dx &= \left[ \frac{1}{3}x^3 - x^2 + 4x ight]_{-1}^2 \
&= \left( \frac{1}{3}(2)^3 - (2)^2 + 4(2) ight) - \left( \frac{1}{3}(-1)^3 - (-1)^2 + 4(-1) ight) \
&= \left( \frac{8}{3} - 4 + 8 ight) - \left( -\frac{1}{3} - 1 - 4 ight) \
&= \frac{20}{3} + \frac{14}{3} \
&= \frac{34}{3} \approx 11.33
\end{align*}

Therefore, the area under the graph of $f(x)$ on $[-1,2]$ is approximately $\frac{34}{3}$ units. The closest answer choice to this value is C) 12.

Find the Area of the Region Under the Graph Of f