Giải thích các bước giải:

Câu 10:

Đáp án: A

$2x^{2} + x + 1 = x^{2} + 3$

$\Leftrightarrow x^{2} + x - 2 = 0 \Leftrightarrow \left[\begin{aligned} &x = 1 \\ &x = -2 \end{aligned}\right.$

$\displaystyle S = \int_{-2}^{1} |(2x^{2} + x + 1) - (x^{2} + 3)| dx = \int_{-2}^{1} |x^{2} + x - 2| dx$

$\displaystyle S = \int_{-2}^{1} (2 - x - x^{2}) dx = \left. \left( 2x - \dfrac{x^2}{2} - \dfrac{x^{3}}{3} \right) \right|_{-2}^{1}$

$\displaystyle S = \left( 2(1) - \dfrac{1^{2}}{2} - \dfrac{1^{3}}{3} \right) - \left( 2(-2) - \dfrac{(-2)^{2}}{2} - \dfrac{(-2)^{3}}{3} \right) = \dfrac{7}{6} - \left(-\dfrac{10}{3}\right) = \dfrac{9}{2}$

Câu 11:

Đáp án: B

$\displaystyle \int_{-2}^{-1} f(x)dx = -S_1$

$\displaystyle \int_{1}^{2} f(x)dx = -S_2$

$\displaystyle \int_{-1}^{1} f(x)dx = S_3$

$\displaystyle \int_{-2}^{2} f(x)dx = \int_{-2}^{-1} f(x)dx + \int_{-1}^{1} f(x)dx + \int_{1}^{2} f(x)dx = -S_1 + S_3 - S_2$

$\displaystyle \int_{-2}^{2} f(x)dx = -\dfrac{22}{15} + \dfrac{76}{15} - \dfrac{22}{15} = \dfrac{32}{15}$