Definite Integral with calculator?

Question

In the textbook it says "By using a calculator/computer we find........ to be ". Is this possible using a TI-30 (we're restricted to this during the test) calculator using a large number of subdivisions?

Answer 1

As a teacher that quite often doesn't allow calculators on exams, I would say that any problem that says to use the caluclator to solve will be worded differently for a test. The book may have been refering to a graphing calculator. Ask your teacher what you would be expected to do with the calculator on the test.

Answer 2

Integration By Substitution Calculator

Answer 3

integral
This article deals with the concept of an integral in calculus. For other meanings of "integral" see integration and integral (disambiguation).
Topics in calculus
Fundamental theorem
Limits of functions
Continuity
Vector calculus
Tensor calculus
Mean value theorem

Differentiation
Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
Table of derivatives

Integration
Lists of integrals
Improper integrals
Integration by: parts, disks,
cylindrical shells, substitution,
trigonometric substitution

In calculus, the integral of a function is an extension of the concept of a sum, typically with direct physical interpretation pertaining to area, mass, or volume. The process of finding integrals is integration, in its mathematical meaning. There are several possible definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.

The term "integral" may also refer to antiderivatives. Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.

Intuitively, the integral of a continuous, positive real-valued function f of one real variable x between a left endpoint a and a right endpoint b represents the area bounded by the lines x = a, x = b, the x-axis, and the curve defined by the graph of f. More formally, if we let


then the integral of f between a and b is the measure of S.

Leibniz introduced the standard long s notation for the integral. The integral of the previous paragraph would be written . The ∫ sign represents integration, a and b are the endpoints of the interval, f(x) is the function we are integrating known as the integrand, and dx is a notation for the variable of integration. Historically, dx represented an infinitesimal quantity, and the long s stood for "sum". However, modern theories of integration are built from different foundations, and the notation should no longer be thought of as a sum except in the most informal sense.

As an example, if f is the constant function f(x) = 3, then the integral of f between 0 and 10 is the area of the rectangle bounded by the lines x = 0, x = 10, y = 0, and y = 3. The area is the width of the rectangle times its height, so the value of the integral is 30. The same result can be found by dividing the area into thin strips and using the Riemann sum or Riemann integral methods[1]. Try it yourself by clicking here -> [2]

Integrals can be taken over regions other than intervals. In general, the integral over a set E of a function f is written . Here x need not be a real number, but, for instance, a vector in R3. Fubini's theorem shows that such integrals can be rewritten as an iterated integral. In other words, the integral can be calculated by integrating one coordinate at a time.


The integral between a and b of f(x) is the area between the curve y = f(x) and the x-axis in the interval [a, b].If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. Integrals result in a number, not another function. If the domain of the function to be integrated is the real numbers, and if the region of integration is an interval, then the greatest lower bound of the interval is called the lower limit of integration, and the least upper bound is called the upper limit of integration.


Finding the area between two curves.
Computing integrals
The most basic technique for computing integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this:

Choose a function f(x) and an interval [a,b].
Find an antiderivative of f, that is, a function F such that F' = f.
By the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration, .
Therefore the value of the integral is F(b) − F(a).
Note that the integral is not actually the antiderivative, but the fundamental theorem allows us to use antiderivatives to evaluate definite integrals.

The difficult step is finding an antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include:

Integration by substitution
Integration by parts
Integration by trigonometric substitution
Integration by partial fractions
Even if these techniques fail, it may still be possible to evaluate a given integral. The next most common technique is residue calculus, whilst for nonelementary integrals Taylor series can sometimes be used to find the antiderivative. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral.

Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.

Specific results which have been worked out by various techniques are collected in the list of integrals.


Approximation of definite integrals
Definite integrals may be approximated using several methods of numerical integration. One popular method, called the rectangle method, relies on dividing the region under the function into a series of rectangles and finding the sum. Other well-known methods are the trapezoidal rule and Simpson's rule.

Some integrals cannot be found exactly, and others are so complex that finding the exact answer would be extremely time-consuming or computationally-intensive. Approximation, however, is a process which relies only on variable substitution, multiplication, addition, and division. It can be done easily and quickly by modern graphing calculators and computers. Many real-world applications of calculus rely on calculating integrals approximately because of the complexity of formulas and since an exact answer is unnecessary.

Answer 4

Yes, but it would take forever to do.