Question

Problem 1 Evaluate J j (sin () (Hint: change the order of integration and combine into one integral) drdy (sin( drdy

Answer 1

integral^1_0 integral^suqareroot 3/2_0 (sin^-1 x)^2 dx dy + integral^2_1 integral^squareroot 3/2_squareroot 1 - y^2 (sin^-1 x)^2 dx dy given that we have change the order of above integral given squareroot 3/2 lessthanorequalto x lessthanorequalto squareroot 1 - 1/y^2 and 1 lessthanorequalto y lessthanorequalto 2 standard the graph of the above in equality shaded region is our concern . from the graph is is clear that 0 lessthanorequalto x lessthanorequalto squareroot 3/2 1 lessthanorequalto y lessthanorequalto 1/squareroot 1 - x^2 x = squareroot 1 - 1/y^2 implies x^2 = 1 1 - 1/y^2 implies x^2 -1 = -1/y^2 = 1 - x^2 = 1/y^2 implies y = 1/squareroot 1 - x^2 that I = integral^1_0 integral^squareroot_0 (sin^-1 x)^2 dx dy + integral^2_0 integral^squareroot 3/2/squareroot 1 - 1/y^2 (sin^-1 x)^2 dx dy = integral^squareroot 3/2_0 integral^1_0 (sin^-1 x)^2 dy dx + integral^squareroot 3/2_0 integhral^1/squareroot 1 - x^2_0 (sin^-1 x)^2 dy dx = integral6 squareroot 3/2_0 { integral