SOLUTION: Find the area or the region bounded by the curves y = x^3 and y = x.

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Question 1085535: Find the area or the region bounded by the curves y = x^3 and y = x.
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
+graph%28400%2C400%2C+-2%2C2%2C+-2%2C2%2C+x%5E3%2C+x%29++
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Noting that the curves cross at x=0 and x=1, with x%3E=x%5E3+ on that interval, and by symmetry, the total area bounded is twice the area of the bounded region in the first quadrant, we can integrate from 0 to 1 and then just double it.
+Area+=+2+%2A+int%28+%28x-x%5E3%29%2C+dx%2C+0%2C+1+%29+
+Area+=+2%2A+%28%281%2F2%29x%5E2+-+%281%2F4%29x%5E4%29+ <<< evaluated at 1 and 0
+Area+=+2+%2A%281%2F2+-+1%2F4+-+%280+-+0%29%29+=+2%281%2F4%29+=+highlight%281%2F2%29+
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For v(t) = −t^2 + 6, the displacement on t=0 to t=6 is:
+int%28%28-t%5E2+%2B+6%29%2C+dt%2C+0%2C+6%29+
= ++-%281%2F3%29t%5E3+%2B+6t+ evaluated at t=6 and t=0
= +%28-6%5E3%29%2F3+%2B+6%2A6+-+%28-0+%2B+0%29+
= +-72+%2B+36+=+highlight_green%28-36ft%29+
The interpretation of -36ft means the displacement is "to the left" assuming you are using the usual convention of positive displacement/velocity/accelerations toward the right, along the positive x-axis.