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

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 .

in progress 0
Vivian 2 weeks 2021-09-10T13:53:52+00:00 2 Answers 0

Answers ( )

    0
    2021-09-10T13:54:54+00:00

    Answer:

    A = 7.

    Step-by-step explanation:

    0
    2021-09-10T13:55:43+00:00

    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-\dfrac23x^3-\dfrac12x^2\right)\bigg|_0^{1/2}=2\left(\dfrac12-\dfrac1{12}-\dfrac18\right)=\boxed{\dfrac7{12}}

    so that A = 7.

Leave an answer

Browse
Browse

27:3+15-4x7+3-1=? ( )