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Applied calculus topic summary: the integral 
Antiderivatives
Antiderivative in words: 
Examples


Indefinite Integral
The expression
is read "the indefinite integral of f(x) with respect to x," and stands for the set of all antiderivatives of f. Thus, ∫ f(x) dx is a collection of functions; it is not a single function, nor a number. The function f that is being integrated is called the integrand, and the variable x is called the variable of integration. (The expression dx is short for "with respect to x.") 
Examples
The constant of integration, C, reminds us that we can substitute any number for C and get a different antiderivative. 

Power Rule for the Indefinite Integral
In Words: Notes

Examples


Indefinite Integrals of Some Exponential and Trig Functions


Some Rules for the Indefinite Integral
(a) Sum and Difference Rules
In Words: (b) Constant Multiple Rule
In Words: Why are these rules true? Because the derivative of a sum is the sum of the derivatives, and similarly for differences and constant multiples. 
Examples
Want some practice? Try the tutorial or exercises. 

Substitution
If u is a function of x, we can use the following formula to evaluate an integral.
Using the Formula Use of the formula is equivalent to the following procedure: 1. Write u as a function of x.
Deciding What to Use for u There is no hard and fast rule, but some guidelines that sometimes work are the following.

Example
To evaluate _{}(x^{2}+1)(x^{3} + 3x  2)^{2} dx, proceed as follows.
Now substitute in the integral to obtain the solution, as follows:
Want some practice? Try the tutorial or exercises. 

Applying the Indefinite Integral: Motion in a Straight Line
If s(t) represents position at time t, then velocity is given by v(t) = s'(t) and acceleration by a(t) = v'(t). This means that
s(t) = _{}v(t) dt. Moreover, for motion due to gravity close to the earth's surface, ignoring air resistance, a(t) ≈ 32 ft/s^{2} is constant. Integrating this twice gives the equations
s(t) = s_{0} + v_{0}t 16t^{2} where v_{0} is the initial velocity and s_{0} is the initial position. 

The Definite Integral as a Sum: Numerical Approach
If u is a function of x, we can use the following formula to evaluate an integral. Riemann Sum If f is a continuous function, the left Riemann sum with n equal subdivisions for f over the interval [a, b] is defined follows. First, partition the interval [a, b] into n equal parts:
x_{0} = a, x_{1} = a + Δx, x_{2} = a + 2Δx, ... x_{n} = a + nΔx = b Next, add up the n products f(x_{0})Δx, f(x_{1})Δx, f(x_{2})Δx, ..., f(x_{n 1})Δx, to get the Riemann sum. Thus,
The Definite Integral If f is a continuous function, the definite integral of f from a to b is defined as
In Words: The function f is called the integrand, the numbers a and b are the limits of integration, and the variable x is the variable of integration. Approximating the Definite Integral To approximate the definite integral, we use a Riemann sum with a large number of subdivisions. 
Example
Let us compute the Riemann sum for the integral_{}_{1}^{1}(x^{2}+1) dx using n = 5 subdivisions. First, to compute the subdivisions:
x_{0} = a = 1 x_{1} = a + Δx = 1 + 0.4 = 0.6 x_{2} = a + 2Δx = 1 + 2(0.4) = 0.2 x_{3} = a + 3Δx = 1 + 3(0.4) = 0.2 x_{4} = a + 4Δx = 1 + 4(0.4) = 0.6 x_{5} = b = 1 The Riemann sum we want is
= [f(1) + f(0.6) + f(0.2) + f(0.2) + f(0.6)]0.4 We can organize this calculation in a table as follows.
The Riemann sum is therefore
If you have Excel and want to see a visual representation of Riemann sums like the picture on the left, download the Excel Riemann Sum Grapher. 

The Definite Integral as Area: Geometric Approach
Geometric Interpretation of the Definite Integral (NonNegative Functions) If f(x) ≥ 0 for all x in [a, b], then _{}_{a}^{b}f(x) dx is the area under the graph of f over the interval [a, b], as shaded in the figure. Geometric Interpretation of the Definite Integral (All Functions) For general functions, _{}_{a}^{b}f(x) dx is the area between x = a and x = b that is above the xaxis and below the graph of f, minus the area that is below the xaxis and above the graph of f. 
Example
Relationship between Riemann Sum Definition and Area Definition The following figure illustrates the relationship between the (left) Riemann sum and the area for the integral _{}_{0}^{1} (1x^{2}) dx. If Δx is the width of each rectangle, then:
Area of the first (leftmost) rectangle = height × width = f(0)Δx = f(x_{0})Δx
Adding the areas of all the rectangles together gives the Riemann Sum. As the number n of rectangles gets larger (so that each of them has width approaching zero) the area represented by the Reimann sum gets closer and closer to the actual area. Thus,


The Definite Integral: Algebraic Approach and the Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (FTC) Let f be a continuous function defined on the interval [a, b]. Then: (a) If A(x) = _{}_{a}^{x} f(t) dt, then A'(x) = f(x), i.e., A is an antiderivative of f, and (b) If f is any continuous antiderivative of f, and is defined on [a, b], then
Part (b) in words: 
Example
Example of (a)
Example of (b)
Another Example of (b)
