AP Calculus AB - Chapters 6 and 7

Differential equation

Equation that involves x, y, and derivatives of y.

Slope fields

Use short line segments to represent the tangent lines of the solution to the differential equation at many different points.

How to draw:

1. Identify and plug x and y values of indicated points into given derivative equation

2. Draw slopes on each indicated points based on answer of derivative equation

Solving differential equations using separation of variables

1. Move everything “y” to left (including dy) and everything “x” to right (including dx). Keep constants on right if possible

2. Integrate both sides, add C to “x” side

3. If given point, plug in and solve for C, then sub value found for C

4. Solve equation for y

Exponential growth and decay

If y changes at a rate proportional to amount present (dy/dx=ky) and y = y₀ when t=0, then y = y₀e^kt where k is the proportional constant.

Exponential growth occurs when k > 0 and exponential decay occurs when k < 0.

Finding area between two curves

Area = ∫[a,b] |f(x) - g(x)| dx

If function is written in terms of x, write the integral as top - bottom; if written in terms of y, write as right - left.

Finding area between two intersecting graphs

1. Find intersecting points by equating functions and making them equal to 0, solve for x or y depending on terms

2. Integrate top - bottom or right - left

Volume of disk

= (area of disk)(width of disk)

= (piR²)(w)

R = radius, w = width

Axes of Revolution formulas (Disk Method)

Horizontal: ∫[a,b] pi(r(x))² dx

Vertical: ∫[c,d] pi(r(y))² dy

Using Disk Method: x-axis

1. Draw graph of given function and conditions to visualize

2. Determine R(x) = top - bottom

3. Integrate using Horizontal Axis of Revolution formula to find the volume

Using Disk Method: y-axis

1. Draw graph of given function and conditions to visualize

2. Determine R(y) = right - left

   1. If given function is in terms of y, rewrite in terms of x

3. Integrate using Vertical Axis of Revolution formula to find the volume

Washer Method

pi∫[a,b] ([R(x)]² - [r(x)]²) dx

*Steps to solving problems using this method is similar to that of Disk Method

Shell Method

2pi∫[a,b] r(x)h(x)dx

*Steps to solving problems using this method is similar to that of Disk and Washer Methods

Volume of solids w/ known cross sections

1. For cross sections of area A(x) taken perpendicular to the x-axis:
   V=∫[a,b] A(x)dx

2. For cross sections of area A(y) taken perpendicular to the y-axis:
   V=∫[c,d] A(y)dx

*Steps to solving problems using this is similar to that of Disk, Washer, and Shell Methods

Volume of square cross sections

V=∫[a,b] (s(x))²dx

Volume of rectangular cross sections

V=∫[a,b] s(x)(height)dx

Volume of quarter circle cross sections

V=pi/4∫[a,b] (s(x))²dx

Volume of semicircle cross sections

V=pi/8∫[a,b] (s(x))²dx

Volume of isosceles right triangle cross section (base is short leg)

V=1/2∫[a,b] (s(x))²dx

Volume of isosceles right triangle cross section (hypotenuse is base)

V=1/4∫[a,b] (s(x))²dx

Volume of equilateral triangle cross section

V=sqrt3/4∫[a,b] (s(x))²dx