All Flashcards
Define definite integral.
The area between a curve and the x-axis over a specified interval.
What is an indefinite integral?
A function that represents the antiderivative of another function, including a constant of integration (+C).
Define Riemann Sum.
An approximation of the area under a curve by dividing it into rectangles or trapezoids.
Define integrand.
The function being integrated.
What is an antiderivative?
A function whose derivative is the given function.
Define accumulation function.
A function defined as the integral of another function, representing the accumulated change of that function.
Define U-Substitution
A technique used to simplify integrals by substituting a function with a new variable, u.
Define Improper Integral
An integral with infinite limits of integration or a discontinuity within the interval of integration.
What does it mean for an improper integral to converge?
The limit of the improper integral exists and is a real number.
What does it mean for an improper integral to diverge?
The limit of the improper integral does not exist or is infinite.
What are the differences between definite and indefinite integrals?
Definite: has bounds, results in a number. | Indefinite: no bounds, results in a function + C.
Compare Left, Right, and Midpoint Riemann Sums.
Left: uses left endpoint for height. | Right: uses right endpoint for height. | Midpoint: uses midpoint for height.
Compare overestimation and underestimation of Riemann Sums.
Overestimate: Area approximation is greater than the actual area. | Underestimate: Area approximation is less than the actual area.
What are the differences between relative and absolute extrema?
Relative: Local max/min within an interval. | Absolute: Overall max/min over the entire domain.
What are the differences between convergent and divergent improper integrals?
Convergent: The limit exists and is a real number. | Divergent: The limit does not exist or is infinite.
Explain accumulation of change.
The net change in a function's value over an interval, found by integrating its rate of change (derivative).
Explain the relationship between integrals and derivatives.
Integration and differentiation are inverse operations; one 'undoes' the other.
Explain how to find units for accumulation problems.
Multiply the units of the y-axis by the units of the x-axis.
What does the sign of the area represent when evaluating integrals?
Area above the x-axis is positive, and area below the x-axis is negative, considering the direction of integration.
How does increasing the number of subintervals affect Riemann Sums?
Increasing the number of subintervals generally leads to a better approximation of the area under the curve.
What are inflection points?
Inflection points occur where the second derivative changes sign.
When does a Right Riemann Sum overestimate the area?
For increasing functions, a Right Riemann Sum will always lead to an overestimate.
When does a Right Riemann Sum underestimate the area?
For decreasing functions, a Right Riemann Sum will always lead to an underestimate.
What does the numerical bound have on the answer when using the Fundamental Theorem of Calculus?
The numerical bound has no impact on the answer.
What is partial fraction decomposition?
The process of undoing a common denominator so that a rational function can be re-written as the sum of rational functions with linear denominators.