AP Calc: Formulas, Rules, Reciprocals, Derivatives, Inverse Derivatives, Integrals...

b*h

Area of a Rectangle

½*b*h

Area of a Triangle

½*(b1+b2)*h

Area of a Trapezoid

π*r²

Area of a Circle

2*π*r

Circumference of a Circle

(4/3)*π*r³

Volume of a Sphere

4*π*r²

Surface Area of a Sphere

π*r²*h

Volume of a Right Circular Cylinder

(1/3)*π*r²*h

Volume of a Right Circular Cone

????

LEFT Riemann Sum Formula

????

RIGHT Riemann Sum Formula

????

MIDPOINT Riemann Sum Formula

Overapproximates

When a RIGHT riemann sum is INCREASING it…

Underapproximates

When a RIGHT riemann sum is DECREASING it…

Underapproximates

When a LEFT riemann sum is INCREASING it…

Overapproximates

When a LEFT riemann sum is DECREASING it…

????

Trapezoidal Sum Formula

Overapproximates

If a trapezoidal sum is CONC UP it…

Underapproximates

If a trapzoidal sum is CONC DOWN it…

Cosecant (csc0)

Reciprocal of Sine (1/sin0)

Secant (sec0)

Reciprocal of Cosine (1/cos0)

Cotangent (cot0)

Reciprocal of Tangent (1/tan0)

cos(x)

Derivative of sin(x)

-sin(x)

Derivative of cos(x)

sec²(x)

Derivative of tan(x)

sec(x)tan(x)

Derivative of sec(x)

-csc²(x)

Derivative of cot(x)

-csc(x)cot(x)

Derivative of csc(x)

1/(1-u²)^1/2 du/dx

derivative AKA dy/dx if y = arcsin(u) AKA sin^-1(u)

-1/(1-u²)^1/2 du/dx

derivative AKA dy/dx if y = arccos(u) AKA cos^-1(u)

1/(1+u²) du/dx

derivative AKA dy/dx if y = arctan(u) AKA tan^-1(u)

-1/(1+u²) du/dx

derivative AKA dy/dx if y = arccot(u) AKA cot^-1(u)

arctan(u) + c

integral of 1/(1+u²) du

arcsin(u) + c

integral of 1/(1-u²)^1/2 du

derivative of outside*inside*derivative of inside

chain rule memory trick

(low*di-high - high*di-low) / low²

quotient rule trick

first*di-last + last*di-first

product rule trick

lnlxl+ c

integral of 1/x dx AKA x^-1 dx

a^x/lna +c

integral of a^x dx — (“a” means any constant integer)

e^x + c

integral of e^x dx

they equal “1/2e^2x + c” and “1/2sin(2x) + c”

for cases like “integral of e^2x dx” and “integral of sin(2x) dx”

make the answer (-) b/c it began with a “c” named trig function!!!

if you take an integral of trig functions “sinx” “csc²” or “cscx*cotx”, what must you remember?

side²

volume of a solid of square cross sections

length*height

volume of a solid of rectangle cross sections

1/2*(pi)*(diameter/2)²

volume of a solid of semi-circle cross sections

(pi)*(diameter/2)²

volume of a solid of circle cross sections

side²(root3 / 4)

volume of a solid of equilateral triangle cross sections

1/2*leg²

volume of a solid of isosceles right triangle cross sections w/ leg on base

(hypotenuse)² / 4

volume of a solid of isosceles right triangle cross sections w/ hypotenuse on base

A = interval integral of (rightORtop - leftORbottom)

Formula for AREA of a region on a graph

V = “pi” * interval integral of (outer² - inner²)

Formula for VOLUME of a region on a graph

“x =” for the outer/inner equations, and “y =” for the interval. make sure to end the formula with “dy”

Being rotated around a vertical line (incl. the y-axis aka x=0) means…

“y =” for the outer/inner equations, and “x =” for the interval. make sure to end the formula with “dx”

Being rotated around a horizontal line (incl. the x-axis aka y=0) means…

V = “pi * interval integral of ((2 - outer²) - (2 - inner²))” if the line is above/to the right of the original region → OR, V = “pi * interval integral of ((outer² - 2) - (inner² - 2)) if the line is below/to the left of the original region.

if rotated around a line with a value > 0 or < 0, such as “x = 2” or “y = 2”, the volume equation must be…

UNIT CIRCLEEEEEE