Question

Evaluate. Then interpret the result in terms of the area above and/or below the X-axis. 1 / 1 f (x3 - 2x) dx 2 2 17 5 (43 - 2x) dx = (Type an integer or a simplified fraction.)

Answer 1

The integral can be decomposed and written as

`\int ^{(1)/(2)}_{-(1)/(2)}(x^3-2x)dx=\int ^{(1)/(2)}_{-(1)/(2)}x^3dx-2\int ^{(1)/(2)}_{-(1)/(2)}xdx`

Now,

`\int ^{(1)/(2)}_{-(1)/(2)}x^3dx=(x^4)/(4)|^{(1)/(2)}_{-(1)/(2)}=(((1)/(2))^4)/(4)-((-(1)/(2))^4)/(4)`
`=0`

And

`\int ^{(1)/(2)}_{-(1)/(2)}xdx=(x^2)/(2)|^{(1)/(2)}_{-(1)/(2)}=(((1)/(2))^2)/(2)-((-(1)/(2))^2)/(2)=0`

Hence,

`\begin{gathered} \int ^{(1)/(2)}_{-(1)/(2)}(x^3-2x)dx=\int ^{(1)/(2)}_{-(1)/(2)}x^3dx-2\int ^{(1)/(2)}_{-(1)/(2)}x=0-2(0) \\ \end{gathered}`
`\boxed{\therefore\int ^{(1)/(2)}_{-(1)/(2)}(x^3-2x)dx=0}`