How do you find the area of the region bounded above by the parabola y=2-x^2, and below by the line y=-x' ?

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Answer 1 · 2018-05-14T15:28:43

Use the form:

$"Area" = int_a^b f_2(x)-f_1(x) dx, f_2(x) > f_1(x)$

We know that $f_2(x) = 2-x^2$ and $f_1(x) = -x$

$"Area" = int_a^b 2-x^2-(-x) dx$

Find the values of $a$ and $b$ by setting the right sides of the two equations equal:

$-x = 2-x^2$

$x^2-x-2=0$

$(x-2)(x+1)=0$

$x = -1$ and $x = 2$

This means that $a = -1$ and $b = 2$

$"Area" = int_-1^2 2-x^2+x dx$

$"Area" = 2x-1/3x^3+1/2x^2|_-1^2$

$"Area" = 2(2)-1/3(2)^3+1/2(2)^2-(2(-1)-1/3(-1)^3+1/2(-1)^2)$

$"Area" = 9/2$