Improper integrals are like definite integrals, only unbounded from the left and/or right. And yes, some of them have finite values. An unbounded area with a finite size? That's calculus to you!
The trapezoidal rule is yet another way to approximate the area under a curve. Learn about they way it's constructed and how accurate it is comparing the rectangular Riemann sums.
Review_Indefinite_integrals_antiderivatives
Review your understanding of indefinite integrals and antiderivatives with some challenge problems.
Definite_integral_as_the_limit_of_a_Riemann_sum
The definite integral of a function is the limit of a sequence of Riemann sums where the intervals become infinitely small. Learn more about this formal definition.
If f' is the derivative of f, then f is the antiderivative of f'. To find the antiderivative of a function we need to perform some kind of reverse differentiation. Learn about it here.
The fundamental theorem tells us how we can evaluate definite integrals. Now let us put this to use and evaluate some!
Indefinite integrals are the way integral calculus deals with antiderivatives. Gain some practice in finding various indefinite integrals.
Functions_defined_by_integrals
Get comfortable with functions that for any input x, return the area under a curve between a given bound and x. This is an important stop on our journey to relating differential and integral calculus.
Fundamental_theorem_of_calculus_chain_rule
Get better acquainted with the fundamental theorem of calculus, and apply it for a class of seemingly hairy problems.
Riemann sums is the name of a family of methods we can use to approximate the area under a curve. Learn about the different ways and how they are constructed.
Definite integrals behave is specific ways with shared properties. For example, the sum of the definite integrals of a function f from a to b and from b to c is equal to the definite integral of f from a to c. Learn about these properties and use them in order to evaluate integrals.
Review_Definite_integral_basics
Review your understanding of the basics of definite integrals with some challenge problems.
Indefinite_integrals_of_common_functions
Indefinite integrals (or antiderivatives) are really just backward differentiation. Therefore, the indefinite integral of eˣ is eˣ+c, the indefinite integral of 1/x is ln(x)+c, the indefinite integral of sin(x) is -cos(x)+c, and the indefinite integral of cos(x) is sin(x)+c.
Review your understanding of Riemann sums with some challenge problems.
Fundamental_theorem_of_calculus
The fundamental theorem of calculus is definitely one of the most important theorems of all time! It relates differential calculus and the concept of the derivative with integral calculus and the concept of the integral. Learn all about it here!
Definite_integrals_of_piecewise_functions
Evaluate definite integrals of piecewise functions. This is where the definite integral properties come into good use.
Riemann_sums_with_sigma_notation
The larger the number of terms in a Riemann sum, the more accurate it is. But as the number of terms increases, we need better ways to write them down. This is where sigma notation comes in very handy.
Introducing the definite integral. It's basically a way to represent the area under a given curve with left-hand and right-hand bounds.
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