. Applied calculus; principles and applications . = 9. f liiy-h^Ydy 10. I :j—^—r = ^arctana;2 = 1 +^* 2 Jo 4 11. I :i—;—^ = aic tan e^ = arc tan e^ — -r Jo 1 +e2^ Jo 4 ^« r^ • o^ 7. f^l-COS20,^ TT 12. j^ s^nHdB^j^ 2 ^^ = 4* 13. (Kos^edS^ f\±^llde=\- Jo Jo ^ 4 Note. — Considering the areas between the axes and the graphs: IT TT sin a; dx = I cos x dx, where n is positive, 0 Jo TT J sin a; da: = 2 | sin a: dx, where n is positive,0 ^O TT J cos a: da; = 2 | cos x dx, if n is an even integer,0 -^o but = 0, if n is an odd integer. 135. Areas of Curves. — As has been shown, the formu-las in
. Applied calculus; principles and applications . = 9. f liiy-h^Ydy 10. I :j—^—r = ^arctana;2 = 1 +^* 2 Jo 4 11. I :i—;—^ = aic tan e^ = arc tan e^ — -r Jo 1 +e2^ Jo 4 ^« r^ • o^ 7. f^l-COS20,^ TT 12. j^ s^nHdB^j^ 2 ^^ = 4* 13. (Kos^edS^ f\±^llde=\- Jo Jo ^ 4 Note. — Considering the areas between the axes and the graphs: IT TT sin a; dx = I cos x dx, where n is positive, 0 Jo TT J sin a; da: = 2 | sin a: dx, where n is positive,0 ^O TT J cos a: da; = 2 | cos x dx, if n is an even integer,0 -^o but = 0, if n is an odd integer. 135. Areas of Curves. — As has been shown, the formu-las in rectangular coordinates are A = j ydx and A = I xdy. AREAS OF CURVES 215 (a) Let A denote the area between the curves y = f (x)and y = F (x); let a; = OM, dx = PE; then, the variablearea A = PoPP and dA = PPDE = (/ (x) - F (x)) dx; area PoPTiP = A= r\f{x)-F (x)) dx, where the points of intersection are (xo, 2/0) and (xi, 2/1). If the locus oi y = F (x) is the a;-axis and Xo and Xi are aand h, this formula reduces to / S{x)
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