SOLUTION: What is the area of the region bounded by the curves {{{ y = arcSin(x) + x - x^2 }}} and {{{ y = arcSin(x) - x + x^2 }}} in the first quadrant and in the second quadrant?

Algebra ->  Surface-area -> SOLUTION: What is the area of the region bounded by the curves {{{ y = arcSin(x) + x - x^2 }}} and {{{ y = arcSin(x) - x + x^2 }}} in the first quadrant and in the second quadrant?      Log On


   

Question 1184303: What is the area of the region bounded by the curves +y+=+arcSin%28x%29+%2B+x+-+x%5E2+ and +y+=+arcSin%28x%29+-+x+%2B+x%5E2+ in the first quadrant and in the second quadrant?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!

A quick check with Desmos online graphing shows that the region bounded by
+y+=+Sin%5E%28-1%29%28x%29+%2B+x+-+x%5E2+ and +y+=+Sin%5E%28-1%29%28x%29+-+x+%2B+x%5E2+ involves quadrants 1, 2, and 3,
so I will just assume that you just made some kind of transcription error.
First find intersection points of the two curves.
===> Sin%5E%28-1%29%28x%29%2Bx-x%5E2+=+Sin%5E%28-1%29%28x%29-x%2Bx%5E2 ===> x = 0, 1.
Over [0,1], Sin%5E%28-1%29%28x%29%2Bx-x%5E2+%3E=+Sin%5E%28-1%29%28x%29-x%2Bx%5E2, and so the area of the region between them is given by



Over [-1,0], Sin%5E%28-1%29%28x%29%2Bx-x%5E2+%3C=+Sin%5E%28-1%29%28x%29-x%2Bx%5E2, and so the area of the region between them is given by

.

Therefore the combined area of the region between the two curves is 5%2F3+%2B+1%2F3+=+highlight%282%29.