Estimate the value of k(0<k<1) so that the region enclosed by y=1/ sqrt(1-x^(2)) ,y=x, x=0, and x=k has an area of 1 square unit.

1 Answers

Best Answer

Set up integral for the area and equate it to 1
int_(0)^(k)[ (1)/(sqrt(1-x^(2)))-x]dx=1
Integrate
[ arcsin (x)- (x^(2))/(2)]_(0)^(k)=1
Evaluate and solve using CAS
(arcsin (k)- (k^(2))/(2))-(arcsin (0)- (0^(2))/(2))=1
arcsin (k)- (k^(2))/(2)=1
arcsin (k)- (k^(2))/(2)-1=0
k~ 0.9973
Result:
k~ 0.9973