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Riemann Sum
Approximation method using rectangles for integrals.
Definite Integral
Integral with specified upper and lower limits.
Midpoint Riemann Sum
Uses midpoints of intervals for rectangle heights.
Left Riemann Sum
Uses left endpoints of intervals for rectangle heights.
Right Riemann Sum
Uses right endpoints of intervals for rectangle heights.
Trapezoidal Approximation
Uses trapezoids for more accurate area approximation.
Area of Trapezoid
A = h(b₁ + b₂)/2, where h is height.
Subdivisions
Dividing interval into smaller segments for approximation.
Volume of Balloon
V = (4/3)πr³, for spherical balloons.
Rate of Volume Change
Volume change per unit time, e.g., cm³/sec.
Infinite Subdivisions
Approaching perfect accuracy in Riemann Sums.
Summation Notation
Σ notation used for summing series or sequences.
Height of Rectangle
Defined by function value at specific points.
Width of Rectangle
Equal width determined by total interval divided by n.
Trapezoidal Rule
More accurate than rectangles for approximating areas.
Function Approximation
Estimating function values using discrete data points.
Example Calculation
Demonstrating methods with specific numerical values.
Limit of Riemann Sums
As n approaches infinity, accuracy improves.
Data Table
Used for approximating areas under curves.
Graphical Representation
Visualizing functions and approximations with sketches.
Equal Width Rectangles
Rectangles of the same width for approximation.
Height Averaging
Using average heights from left and right sums.
Setup for Riemann Sums
Clearly show calculations for approximating integrals.
tan⁻¹(x)
Inverse tangent function, returns angle whose tangent is x.
arcsin(1)
Returns angle where sine equals 1, equals π/2.
sin⁻¹(x)
Inverse sine function, returns angle whose sine is x.
arctan(1)
Returns angle where tangent equals 1, equals π/4.
Long Division
Method to simplify fractions before integration.
Second Fundamental Theorem
Relates differentiation and integration for continuous functions.
Chain Rule
Method for differentiating composite functions.
Logarithmic Integration
Integrating functions using natural logarithm properties.
Integration by Substitution
Technique to simplify integrals using variable change.
Integration of cot(x)
Results in ln|sin(x)| + C.
Integration of tan(x)
Results in -ln|cos(x)| + C.
Integration of arcsin(u)
Results in arcsin(u) + C.
Integration of arctan(u)
Results in arctan(u) + C.
Integration of sec²(x)
Results in tan(x) + C.
Integration of 1/(a²+u²)
Results in (1/a)arctan(u/a) + C.
Integration of 1/√(a²-u²)
Results in arcsin(u/a) + C.
Integration of 1/(1+u²)
Results in arctan(u) + C.
Differentiation of ln|x|
Results in 1/x.
Integration of x²
Results in (1/3)x³ + C.
Integration of sec(x)tan(x)
Results in sec(x) + C.
Integration of sin(x)
Results in -cos(x) + C.
Integration of cos(x)
Results in sin(x) + C.
Definite Integral
Integral with specific upper and lower limits.
Signed Area
Area between curve and x-axis, can be positive or negative.
Fundamental Theorem of Calculus
Links differentiation and integration; f(b) - f(a).
Calculator Integration
Using TI-84 to compute definite integrals.
Indefinite Integral
Represents a family of curves, includes constant of integration.
Continuous Function
Function without breaks or discontinuities on an interval.
Velocity Equation
Describes the rate of change of position over time.
Displacement
Net change in position over a specified interval.
Antiderivative
Function whose derivative gives the original function.
Integration Syntax
Specific format for using fnInt on TI calculators.
Graphical Representation
Visual display of the area under a curve.
Area Under Curve
Total area between the curve and x-axis.
Initial Value Problem
Differential equation with specified initial conditions.
Position Equation
Describes an object's location as a function of time.
Definite Integral Evaluation
Calculating the integral value between two limits.
MATHPRINT Setting
Calculator mode for simplified display of calculations.
Continuous Function on [a, b]
Function remains uninterrupted between limits a and b.
Signed Area Calculation
Determining area considering direction above or below x-axis.
Function Value at Endpoint
Calculated using initial value plus definite integral.
Velocity Function
Derivative of position function with respect to time.
Integration by Parts
Technique for integrating products of functions.
Midpoint Rule
Approximation method using midpoints for area under curve.
Constant of Integration
Arises in indefinite integrals, represents family of solutions.
Antiderivative
Function whose derivative gives the original function.
Integration Rules
Set of rules for finding antiderivatives.
Constant of Integration
Arbitrary constant added to antiderivatives.
Power Rule
Integrate x^n as (x^(n+1))/(n+1) + C.
Sum Rule
Integral of sum equals sum of integrals.
Constant Rule
Integral of constant k is kx + C.
Exponential Rule
Integral of e^x is e^x + C.
Trigonometric Rules
Specific integrals for trig functions.
u-Substitution
Method for simplifying integrals using substitution.
Derivative of e^x
Derivative is e^x, same as original.
Derivative of sin x
Derivative is cos x.
Derivative of cos x
Derivative is -sin x.
Derivative of sec x
Derivative is sec x tan x.
Integral of sin x
Integral is -cos x + C.
Integral of cos x
Integral is sin x + C.
Integral of sec² x
Integral is tan x + C.
Integral of csc² x
Integral is -cot x + C.
Integral of sec x tan x
Integral is sec x + C.
Integral of csc x cot x
Integral is -csc x + C.
Initial Condition
Value used to find constant of integration.
Velocity Equation
Derived from acceleration by integrating.
Position Equation
Derived from velocity by integrating.
Common Errors
Sign errors in trigonometric functions.
Antiderivative of sec² x
tan x + C.
Antiderivative of csc x cot x
-csc x + C.
Antiderivative of sin² x
(1/2)(x - (1/2)sin(2x)) + C.
Antiderivative of cos² x
(1/2)(x + (1/2)sin(2x)) + C.
Integration by Parts
Method using u and dv to integrate.
Note 
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