Basic Integration Formulas and the Substitution Rule 1The second fundamental theorem of integral calculus Recall fromthe last lecture the second fundamental theorem ofintegral calculus. Theorem Let f(x) be a continuous function on the interval a,b. Let F(x) be any function withthe property that F (x) = f(x) Then ∫b a f(x)dx = F(b) - F(a. Integrals 4.jpg - 6 Basic Antidifferentiation /Integration General and Particular Solutions Solutions found on Teacher Page under AP Calculus AB.
1. The Indefinite Integral
Recall that if I is any interval and f is a function definedon I, then a function F on I is anantiderivative of f if F'(x) = f(x) for all x ∈ I. A corollary ofthe Mean Value Theorem tells us that if G is any other antiderivative of f on I, then there is a constant C such that G(x) = F(x) + C.
Let I be an interval and f a function defined on I.Then the indefinite integral of f is defined tobe the set of all antiderivatives of f on I. We denote this set by:
We call f the integrand and x the variable ofintegration.
Since all the antiderivatives of f differ only by a constant, it is customary to write:
where F is any particular antiderivative, and C is the constant of integration which implicitly takes on all real number values.
2. The Definite Integral
Let f be a function defined on a closed interval [a, b]. Given any n we let Δx=(b-a)/n, and we let xk = a + k Δx. For each k, select sample pointsxk* ∈[xk-1, xk]. Wedefine the definite integral of f on [a, b]to be
provided this limit exists and does not depend on the choice of sample points xk*.
If the limit depends upon the choice of sample points, or is undefined forany choice of sample points, then the integral does not exist.
6 Basic Integration & Applicationsap Calculus 14th Edition
We call a the lower limit of theintegral, and b the upper limit. We callf the integrand.
We also make the defintions:
The integral sign '∫' is an archaic 'S' and stands for 'sum' (as does ∑, of course).
If it exists, the definite integral gives you a number as its result. In this regard, x is a 'dummy variable' and the dx reminds us what variable we are integrating with respect to. We can just as easily replace x with t and write:
and the numerical value of the definite integral is unchanged.
Geometrically, if f(x) ≥ 0 we interpret the integral as thearea between the graph y=f(x), the x-axis, andthe lines x=a and x=b. More generally,we think of the integral as a 'signed' area, where the integral is equal tothe area above the x-axis less the area below the x-axis.
6 Basic Integration & Applicationsap Calculus Solver
in the definition of the definite integral is called a Riemann sum; the definite integral is sometimes called a Riemann integral (to distinguish it from other more general integrals used by mathematicians). Whenever you have a limit as
For certain simple functions, you can calculate an integral directly usingthis definition. However, in general, you will want to use the fundamentaltheorem of calculus and the algebraic properties of integrals.
3. Rules of Integration
Let f and g be functions and let a, b and c be constants, and assume that for each fact all the indicated definite integrals exist. Then the following are true:
Constants can be pulled out of integrals:
The integral of the sum of two functions equals the sum of the integrals of each function:
The integral of the difference of two functions equals the difference of the integrals of each function:
The integral from a to b of a function equals the integral from a to c plus the integral from c to b:
Note that there are no general rules for integrals of products andquotients. Such integrals can sometimes, but not always, be calculated usingsubstitution or integration by parts.
We can also give some facts about inequalities involving definite integrals.
If f(x) ≥ 0 on [a, b] then
If f(x) ≤ g(x) on [a, b] then
The absolute value of an integral is less than the integral of the absolute value.