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.
How to find where h(x) is increasing?
Find where the graph is positive, which means above the x-axis.
How to find where h(x) is concave up?
Find where the slope of f(x) is positive because that would be a positive second derivative of f(x).
How to find inflection points for h(x)?
Look for slope changes in the graph of f(x).
How to find relative extrema for h(x)?
Use the derivative graph, f(x), to find changes in sign of the first derivative.
How to find absolute extrema for h(x)?
Identify candidates for absolute maximum and absolute minimum for the integrally defined function.
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.