. Applied calculus; principles and applications . If the initial ordinate from which area is reckoned isundetermined, then / V2 ax — x^ dx X — a 1 ^ V2ax-x + ^a2 sin-i -—- + C,2 2 a 7ra^ where C = ^ , if Area = 0, when x = 0; or X — a ^ V2ax - x + ^a2vers-i- + C where C = 0, if Area = 0, when a: = may be seen in the figure, sm- x — a , TT + ;i = OCP = vers-i -; a 2 a that is, a^ / . .x — a , Tr\ a^ .2K^ + 2J=2^ x — a . ira^ a^ 4--r = Trvers^-a 4 2 a (See Note at end of Exercise XXIII.)Either result gives ^2 ax — x^dx = \Tra^ = areaof 05A. AREA UNDER EQUILATERAL HYPERBOLA 221 Example 3. —
. Applied calculus; principles and applications . If the initial ordinate from which area is reckoned isundetermined, then / V2 ax — x^ dx X — a 1 ^ V2ax-x + ^a2 sin-i -—- + C,2 2 a 7ra^ where C = ^ , if Area = 0, when x = 0; or X — a ^ V2ax - x + ^a2vers-i- + C where C = 0, if Area = 0, when a: = may be seen in the figure, sm- x — a , TT + ;i = OCP = vers-i -; a 2 a that is, a^ / . .x — a , Tr\ a^ .2K^ + 2J=2^ x — a . ira^ a^ 4--r = Trvers^-a 4 2 a (See Note at end of Exercise XXIII.)Either result gives ^2 ax — x^dx = \Tra^ = areaof 05A. AREA UNDER EQUILATERAL HYPERBOLA 221 Example 3. — Find / (mx + h) dx, by means of line= mx + h. Area = OMPB = BDP + OMDB= ix mx -\- Xb mx^ hx. If the initial ordinate is riot OB, and is undetermined, then / mx mx -\-b) dx = -^ + 6a; + 137. Area under Equilateral Hyperbola. — As in the figure of the circle y = Va^ — x^, f Va^ — X? dx = -^x Va^ — x^ + ^ a^ sin~i X . expresses the area BOMP and a^ sin^ - is represented bytwice the area of the circular sector BOP] so V^^T^^dx =lx\/^M^2_|_ 1 asinh-i - (Ex. 7, Art. 123)0 ^ Z a may be shown to express the area AOMP under the equi- lateral hyperbola y = vaM-^> and a^sinh-i- to be repre-sented by twice the area of the hyperboUc sector AOP, 222 INTEGRAL CALCULUS To getI Va^ +x^dx; let x = asinh^, dx = acosh0d0; then, / ^ VoHP^ dx = a2 fcosh^ 0 d0 = |- (</) + sinh (^ cosh 0) = io; Va^ + x2 +1 a^sinh-i -,2 2 a as also in Ex. 7, Art. 123.
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