0 / | | _________ | / 2 2 | \/ x - y dx | / -r
Integral(sqrt(x^2 - y^2), (x, -r, 0))
// 2 /x\ \ || y *acosh|-| 3 | 2| | || \y/ x x*y |x | | ||- ----------- + ------------------- - ----------------- for |--| > 1| || 2 _________ _________ | 2| | / || / 2 / 2 |y | | | || / x / x | | _________ || 2*y* / -1 + -- 2* / -1 + -- | | / 2 2 || / 2 / 2 | | \/ x - y dx = C + |< \/ y \/ y | | || | / || ________ | || / 2 | || / x | || 2 /x\ I*x*y* / 1 - -- | || I*y *asin|-| / 2 | || \y/ \/ y | || ------------ + -------------------- otherwise | \\ 2 2 /
/ 0 | / | | | | / 2 4 2 | 2| | | | y x x 3*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for |--| > 1 | | | _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2\ / 2 |y | | | | / x | x | 3 | x | / x | | | / -1 + -- 2*y*|-1 + --| 2*y *|-1 + --| 2*y* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ y \ y / \ y / \/ y | | | | | | ________ | | < / 2 dx for r > 0 | | | / x | | | I*y* / 1 - -- | | | / 2 2 | | | \/ y I*y I*x | | | ------------------ + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | |/ |-r < | -r | / | | | | / 2 4 2 | 2| | | | y x x 3*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for |--| > 1 | | | _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2\ / 2 |y | | | | / x | x | 3 | x | / x | | | / -1 + -- 2*y*|-1 + --| 2*y *|-1 + --| 2*y* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ y \ y / \ y / \/ y | | | | | | ________ |- | < / 2 dx otherwise | | | / x | | | I*y* / 1 - -- | | | / 2 2 | | | \/ y I*y I*x | | | ------------------ + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | | / \ 0
=
/ 0 | / | | | | / 2 4 2 | 2| | | | y x x 3*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for |--| > 1 | | | _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2\ / 2 |y | | | | / x | x | 3 | x | / x | | | / -1 + -- 2*y*|-1 + --| 2*y *|-1 + --| 2*y* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ y \ y / \ y / \/ y | | | | | | ________ | | < / 2 dx for r > 0 | | | / x | | | I*y* / 1 - -- | | | / 2 2 | | | \/ y I*y I*x | | | ------------------ + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | |/ |-r < | -r | / | | | | / 2 4 2 | 2| | | | y x x 3*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for |--| > 1 | | | _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2\ / 2 |y | | | | / x | x | 3 | x | / x | | | / -1 + -- 2*y*|-1 + --| 2*y *|-1 + --| 2*y* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ y \ y / \ y / \/ y | | | | | | ________ |- | < / 2 dx otherwise | | | / x | | | I*y* / 1 - -- | | | / 2 2 | | | \/ y I*y I*x | | | ------------------ + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | | / \ 0
Piecewise((Integral(Piecewise((-y/sqrt(-1 + x^2/y^2) + x^2/(2*y*(-1 + x^2/y^2)^(3/2)) - x^4/(2*y^3*(-1 + x^2/y^2)^(3/2)) + 3*x^2/(2*y*sqrt(-1 + x^2/y^2)), |x^2/y^2| > 1), (i*y*sqrt(1 - x^2/y^2)/2 + i*y/(2*sqrt(1 - x^2/y^2)) - i*x^2/(2*y*sqrt(1 - x^2/y^2)), True)), (x, -r, 0)), r > 0), (-Integral(Piecewise((-y/sqrt(-1 + x^2/y^2) + x^2/(2*y*(-1 + x^2/y^2)^(3/2)) - x^4/(2*y^3*(-1 + x^2/y^2)^(3/2)) + 3*x^2/(2*y*sqrt(-1 + x^2/y^2)), |x^2/y^2| > 1), (i*y*sqrt(1 - x^2/y^2)/2 + i*y/(2*sqrt(1 - x^2/y^2)) - i*x^2/(2*y*sqrt(1 - x^2/y^2)), True)), (x, 0, -r)), True))
Use the examples entering the upper and lower limits of integration.