Question

Use the given partition and sample points to approximate the definite integral off(x) = x2 + 2x + 2 on the indicated interval.

Answer 1

The given function is

$f(x)=x^2+2x+2$

$\int ^2_{-2}f(x)dx=\int ^2_{-2}(x^2+2x+2)dx$

$=\int ^2_{-2}x^2dx+\int ^2_{-2}2xdx+\int ^2_{-2}2dx$

$=\lbrack\frac{x^3}{3}\rbrack^2_{-2}+\lbrack\frac{2x^2}{2}\rbrack^2_{-2}+\lbrack2x\rbrack^2_{-2}$

$=\frac{(2)^3_{}-(-2)^3}{3}^{}_{}+\frac{2(2)^2-2(-2)^2}{2}+2(2)-2(-2)$

$=\frac{8^{}_{}-(-8)^{}}{3}^{}_{}+\frac{2\times4^{}-2\times4^{}}{2}+4-(-4)$

$=\frac{8^{}_{}+8^{}}{3}^{}_{}+\frac{8-8^{}}{2}+4+4$

$=\frac{16^{}}{3}^{}_{}+^{}0+8$

$=\frac{16+3\times8^{}}{3}^{}_{}$

$=\frac{16+24^{}}{3}^{}_{}$

$=\frac{40^{}}{3}^{}_{}=13.333$

Hence the required value is

$\int ^2_{-2}f(x)dx=\int ^2_{-2}(x^2+2x+2)dx=13.333$