Question 1

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

Question 2

The given function is

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

$=\int ^2_(-2)x^2dx+\int ^2_(-2)2xdx+\int ^2_(-2)2dx$

$=\lbrack(x^3)/(3)\rbrack^2_(-2)+\lbrack(2x^2)/(2)\rbrack^2_(-2)+\lbrack2x\rbrack^2_(-2)$

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

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

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

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

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

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

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

Hence the required value is

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

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