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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: $\int u dv = uv - \int v du$.
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 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.