Calculus
Integral Formulas
[Cartesian Functions] y = f ( x ) , x = g( y )
x2
(1.) A = ∫ f ( x ) dx
x1
y2
(2.) A = ∫ g( y ) dy
y1
x2
(3.) V = ∫ π (f ( x )) dx
(disc method, rotated about x-axis)
(4.) V = ∫ π (g( y )) dy
(disc method, rotated about y-axis)
2
x1
y2
(5.) L = ∫
2
y1
x2
1+ ( dydx ) dx = ∫
y2
2
x1
x2
(6.) S = ∫ 2π y 1+ (
x1
x2
f (x )
(7.) S = ∫ 2π x 1+ (
x1
y1
1+ ( dydx ) dy
2
y2
2
) dx = ∫y 2π y 1+ ( dxdy ) dy (rotated about x-axis)
dy 2
dx
1
y2
2
) dx = ∫y 2πgx(y ) 1+ ( dydx ) dy (rotated about y-axis)
dy 2
dx
1
1 x
(8.) x = ∫ xf ( x ) dx
A x
y=
2
1
1 y 1
2
(f ( x )) dx
∫
y
A
2
2
1
[Parametric Functions] x = x (t ) , y = y (t )
dy dtd ( y ) dydt
d 2y
d ⎛ dy ⎞
= dx = dx
= ⎜⎜ ⎟⎟ =
(9.)
2
dx
dx
dx ⎝ dx ⎠
dt
dt
t
(10.) A = ∫ y (t )x ′(t ) dt
d
dt
( dydx )
dx
dt
2
t1
t2
(11.) A = ∫ x (t )y ′(t ) dt
t1
t2
( x ′(t ))
(12.) L = ∫
2
t1
t2
+ ( y ′(t )) dt
2
2
(13) V = ∫ π ( y (t )) x ′(t ) dt
(disc method, rotated about x-axis)
t1
t2
(14.) V = ∫ π ( x (t )) y ′(t ) dt
2
t1
t2
(disc method, rotated about y-axis)
(15.) S = ∫ 2π y (t )
( x ′(t ))
+ ( y ′(t )) dt
(rotated about x-axis)
(16.) S = ∫ 2π x (t )
( x ′(t ))
+ ( y ′(t )) dt
(rotated about y-axis)
t1
t2
t1
2
2
2
2
[Polar Functions] r 2 = x 2 + y 2 , tanθ =
dy
r cosθ + ddrθ sinθ
=
dx −r sinθ + ddrθ cosθ
θ
(18.) A = ∫ 21 r 2 d θ
(17.)
2
(19.) L = ∫
θ1
θ2
θ1
r 2 + ( ddrθ ) d θ
2
y
, x = r cosθ , y = r sinθ
x