Calculus 2 Formulas and Properties (Cal Poly)

Area Between Curves Top-Bottom

∫a-b of f(x)-g(x) dx

Area Between Curves Right-Left

∫c-d f(y)-g(y) dy

Washer (vertical rectangles)

pi ∫a-b (Outer Radius)^2 - (Inner Radius)^2 dx

Washer (horizontal rectangles)

pi ∫c-d (Outer Radius)^2 - (Inner Radius)^2 dy

Shell (horizontal rectangles)

2pi ∫a-b (radius(x))(height(x)) dx

Shell (vertical rectangles)

2pi ∫c-d (radius(y))(height(y)) dy

d/dx (f inverse (x))

1 / (f prime of (f inverse of x))

Work Formula (W)

∫a-b F(x) dx

Surface Area of an Arc Length Revolved around axis

2pi ∫a-b (radius) √(1+(f'(x))^2) dx

Trig Sub Expression √(a^2 - x^2)

x = a sin (θ); 1 - sin^2(θ) = cos^2(θ)

Trig Sub Expression √(a^2 + x^2)

x = a tan (θ); 1 + tan^2(θ) = sec^2(θ)

Trig Sub Expression √(x^2 - a^2)

x = a sec (θ); sec^2(θ) - 1 = tan^(θ)

∫1/x dx

ln|x| +C

∫e^x dx

e^x +C

∫a^x dx

(a^x)/ln(a) +C

∫x^n

(x^n+1)/n+1 +C

d/dx (ln x)

1/x

d/dx (b^x)

(lnb) * b^x

d/dx (log (base b) x)

1/(lnb)x

d/dx ln(g(x))

g'(x)/g(x)

d/dx (e^g(x))

e^g(x) * g'(x)

log (base a) (A^c)

C * (log (base a) A)

trig identity sin^2(x)+cos^2(x)

1

trig identity tan^2(x)+1

sec^2(x)

cos^2(x)

(1+cos2x)/2

sin^2(x)

(1-cos2x)/2

sin(2x)

2sinxcosx

cos(2x)

cos^2(x)-sin^2(x) = 1-2sin^2(x) = 2cos^2(x)-1

∫du/√(a^2-u^2)

arcsin(u/a) + C

∫du/(a^2+u^2)

(1/a)arctan(u/a)+C

∫cos(x)dx

sin(x)+C

∫sin(x)dx

-cos(x)+C

∫tan(x)dx

ln |sec(x)|+C

∫sec(x)dx

ln |sec(x)+tan(x)|+C

∫csc(x)dx

-ln |csc(x)+cot(x)|+C

∫cot(x)dx

ln |sin(x)|+C

d/dx (sin(x))

cos(x)

d/dx (cos(x))

-sin(x)

d/dx (tan(x))

sec^2(x)

d/dx (csc(x))

-csc(x)cot(x)

d/dx (sec(x))

sec(x)tan(x)

d/dx (cot(x))

-csc^2(x)