integral for range in x axis Calculate the area in the shaded region:
${f(x) = x^3 -2x + 7}$
${{{\int_{-1}^2}} f(x) dx = x^4 - x^2 + 7x}$
$= {[(2^4) -(2^2) + 14] - [(-1)^4 - (-1)^2 + 7(-1)]}$
$= [16 - 4 + 14] - [- 7] = 19$
But the answer in the book is $21{3\over 4}$.


As @Sean said ,you forgot to divide $x^4$ by $4$
Also if you want to find the integral of ${f(x) = x^3 -2x + 7}$
So the result will be $\require{cancel} \cancel{\int_{-1}^{2}} \frac{x^4}{\color{blue}4}-x^2+7x+C$
$$\int_{-1}^{2}\left(x^3-2x+7\right)dx=\left(\frac{x^4}{\color{blue}4}-x^2+7x\right)\bigg|_{-1}^{2}=4-4+14-(1/4-1-7)=\color{red}{21\frac{3}{4}}$$
