Let ( mathrm{R} ) be the region bounded by the functions ( f(x)=-2 x^{2}+1 ) and ( g(x)=-2 x-3 ) as shown in the diagram below. Find the exact area of the region ( mathrm{R} ) without using a calculator. Write your answer in simplest form.
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To do this, we set the two functions equal to each other and solve for x: -2x^2 + 1 = -2x - 3 This simplifies to: 2x^2 - 2x - 4 = 0 Dividing through by 2 gives: x^2 - x - 2 = 0 Factoring this gives: (x - 2)(x + 1) = 0 So the points of intersection are x Show more…
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