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Neko [114]
3 years ago
10

Find the number b such that the line y = b divides the region bounded by the curves x = y^2 − 1 and the y-axis into 2 regions wi

th equal area. Give your answer correct to 3 decimal places.
Mathematics
1 answer:
77julia77 [94]3 years ago
7 0
Because of the symmetry, we can just go from x=0 to x=2 to find the area between 
<span>y = x^2 and y = 4 </span>

<span>that area = ∫4-x^2 dx from 0 to 2 </span>
<span>= [4x - (1/3)x^3] from 0 to 2 </span>
<span>= 8 - 8/3 - 0 </span>
<span>= 16/3 </span>

<span>so when y = b </span>
<span>x= √b </span>
<span>and we have the area as </span>
<span>∫(b - x^2) dx from 0 to √b </span>
<span>= [b x - (1/3)x^3] from 0 to √b </span>
<span>= b√b - (1/3)b√b - 0 </span>

<span>(2/3)b√b = 8/3 </span>
<span>b√b =4 </span>
<span>square both sides </span>
<span>b^3 = 16 </span>
<span>b = 16^(1/3) = 2 cuberoot(2) </span>
<span>or appr 2.52</span>
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