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#