Calculus: Integral basics
Riemann sum subintervals
Determined by: b-a / n so the subintervals are: 0, 0.5, 1, 1.5 (left) 0.5, 1, 1.5, 2 (right)
The ∫ part of an integral means?
summation
The Fundamental Theorem of Calculus
A simplified equation for determining the EXACT area under a continuous function curve.
What is an Indefinite Integral? What is it's other name?
Also known as an antiderivative. When evaluated it will return the original function of the derivative in the integrand. The +c is the constant of integration and is used to account for any arbitrary constant in the original function.
Relationship between the definite integral and the Riemann sum?
Essentially, the Rieman sum sums the rectangles over the interval a to b. This gives an approximation of the area that becomes more accurate as more rectangles are used
What is the shorthand version of the Fundamental Theorem of Calculus
F(x) is the antiderivative. b and a are the upper and lower limits of integration respectively.
For a Riemann Sum what is the height and width of each rectangle?
Height: f(x) the distance from axis to curve. Width: ∆x = b-a / n Distance from a to b (limits of integration) divided by the number of rectangles n.
How are indefinite and definite integrals different?
Indefinite integrals will output a function with an arbitrary + c. Definite integrals are evaluated over a specific interval (has limits of integration) and will output a constant. This constant is the area under the curve.
Substitution Rule for INDEFINITE Integrals?
Let something be u. To simplify part of the integrand. Take du to eliminate the rest of the integrand and dx so that the entire integrand is defined in terms of u.
What is L.I.A.T.E for Integration by parts?
Mnemonic for choosing u and dv. Top should be u and towards the bottom dv. L: Logarithmic functions. I: Inverse trig. A: Algebraic functions T: Trigonometric functions. E: Exponential functions.
Substitution Rule for DEFINITE Integrals?
Similar to indefinite integrals except you must plug your limits of integration into the function that equals u if you decide to keep everything in terms of du. Else convert back to x after evaluating the integral in terms of u.
What is the summation notation for a definite integral?
Summation from rectangle 1 to an infinite number of them, ∆x apart, evaluating each point for f(xi) for area.
Left-hand Riemann sum?
The curve is touched by the left side of each rectangle. Sum from i = 0 to n-1
Right-hand Riemann Sum?
The curve is touched by the right side of each rectangle. Sum from i = 1 to n
What is the dx portion of an integral?
The variable of integration. It does the following: 1) Encloses the entire function we're integrating 2) Indicates what we're integrating with respect to. The axis over which we're working with infinitesimally small increments in the values.
What is the integrand?
What you're integrating
What is true for constants in the integrand?
You can factor them out of the integral and apply them after evaluating.
What is true of the integral of a sum (or difference)?
You can split it into two integrals for easier evaluation
Limits of integration
a = lower limit b = upper limit
The substitution method is typically used to reverse the _______. What is the typical indicator?
chain rule. Useful if it looks like you have an inner and outer layer. Or just a very complex integral in general.
Integration by parts is used to reverse the ______ What is the typical indicator?
product rule. Useful if it looks like you have a function being multiplied by a derivative.