Question

Find the area of the region trapped between LaTeX: y=1-2x^2 y = 1 − 2 x 2 and LaTeX: y=\left|x\right| y = | x | , shown above. The answer is LaTeX: \frac{A}{12} A 12 . Below, enter only the whole number LaTeX: A A .

Answer 1

The area is given by the integral,

`\displaystyle\int_(-1/2)^(1/2)(1-2x^2-|x|)\,\mathrm dx`

The integrand is even, so we can simplify the integral somewhat as

`\displaystyle2\int_0^(1/2)(1-2x^2-|x|)\,\mathrm dx`

When
`x\ge0`, we have
`|x|=x`, so this is also the same as

`\displaystyle2\int_0^(1/2)(1-2x^2-x)\,\mathrm dx`

which has a value of

`2\left(x-\frac23x^3-\frac12x^2\right)\bigg|_0^(1/2)=2\left(\frac12-\frac1{12}-\frac18\right)=\boxed{\frac7{12}}`

so that A = 7.