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(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`