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.
How to approximate area under a curve using Riemann Sums?
Divide the interval into subintervals, create rectangles/trapezoids, calculate their areas, and sum them up.
How to evaluate a definite integral using FTC?
Find the antiderivative of the function, evaluate it at the upper and lower limits, and subtract the values.
How to find the area given a rate of change function?
Integrate the rate of change function over the given interval.
How to find the total distance traveled given a velocity function?
Integrate the absolute value of the velocity function over the given interval.
How to use u-substitution to solve an integral?
Choose u, find du, rewrite the integral in terms of u, integrate, and substitute back to the original variable.
How to find the value of C in an indefinite integral?
Use the initial condition given, substitute into the equation, and solve for C.
How to evaluate a definite integral with u-substitution?
Choose u, find du, change the bounds, rewrite the integral in terms of u, integrate, and evaluate using the new bounds.
How to solve integrals using long division?
Use long division to rewrite the integral, then integrate term by term.
How to solve integrals by completing the square?
Rewrite the denominator by completing the square, then use u-substitution and inverse trig functions.
How to solve integrals using integration by parts?
Choose u and dv, find du and v, and apply the formula: .