Applied calculus topic summary: the integral

Tools: Numerical Integration Utility | Excel Riemann Sum Grapher

Subtopics: Antiderivatives | Indefinite Integral | Power Rule for Indefinite Integral | Integrals of Exponential & Trig Functions | Some Rules for the Indefinite Integral | Substitution | Application: Motion in a Straight Line | Definite Integral as a Sum: Numerical Approach | Definite Integral as Area: Geometric Approach | Definite Integral: Algebraic Approach and the Fundamental Theorem of Calculus

Antiderivatives

An antiderivative of a function f is a function F such that F'= f.

Antiderivative in words:

An antiderivative of a given function is a function whose derivative is the given function.

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Examples

 1 An antiderivative of 4x3 is x4     Because the derivative of x4 is 4x3 2 Another antiderivative of 4x3 is x4 + 7     Because the derivative of x4 + 7 is 4x3 3 An antiderivative of 2x is x2 + 12.     Because the derivative of x2+12 is 2x

 4. An antiderivative of 5 is 5. An antiderivative of x is

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Indefinite Integral

The expression

 f(x) dx

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.")

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Examples

 1 4x3 dx  = x4 + C C is an arbitrary constant 2 2x dx  = x2 + C

The constant of integration, C, reminds us that we can substitute any number for C and get a different antiderivative.

 3 5 dx = 4 x dx =

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Power Rule for the Indefinite Integral
 xn dx = xn+1n + 1 + C if n ≠ -1 x-1 dx = ln|x|+ C

In Words:

To find the integral of xn, add one to the exponent, then divide by the new exponent. This rule works provided n is not -1.

Notes
1.
 The integral 1 dx is commonly written as dx.
2.
 Similarly, 1x55 dx may be written as dxx55 .

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Examples

1.
 x55 dx
=
 x5656 + C
2.
 1x55 dx
=
 x-55 dx = - x-5454 + C
3.
 dx
=
 x0 dx = x11 + C = x + C

 4 x3 dx = 5 1x2 dx =

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Indefinite Integrals of Some Exponential and Trig Functions
 ex dx = ex + C
 Because ddx ex = ex
 cos x dx = sin x + C
 Because ddx sin x = cos x
 sin x dx = -cos x + C
 Because ddx (-cos x) = sin x
 sec2x dx = tan x + C
 Because ddx tan x = sec2x

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Some Rules for the Indefinite Integral

(a) Sum and Difference Rules

 [f(x) ± g(x)] dx = f(x) dx ± g(x) dx

In Words:

The integral of the sum of two functions is the sum of the individual integrals, and the integral of the difference of two functions is the difference of their integrals.

(b) Constant Multiple Rule

 kf(x) dx = k f(x) dx (k constant)

In Words:

To take the integral of a constant times a function, take the integral of the function by itself, and then multiply the answer by that constant. (In other words, the constant "goes along for the ride" )

Why are these rules true? Because the derivative of a sum is the sum of the derivatives, and similarly for differences and constant multiples.

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Examples

Sum:
 (x3 + 1) dx
=
 x3 dx + 1 dx
=
 x44 + x + C
Constant
Multiple:
 5x3 dx
=
 5 x3 dx
=
 5x44 + C
Constant
Multiple:
 4 dx
=
 4 1 dx
=
 4x + C
Both Rules:
 (6x2 + 4) dx
=
 6 x dx + 4 1 dx
=
 6x33 + 4x + C
=
 2x3 + 4x + C

Want some practice? Try the tutorial or exercises.

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Substitution

If u is a function of x, we can use the following formula to evaluate an integral.

 f dx = fdu/dx du

Using the Formula

Use of the formula is equivalent to the following procedure:

1. Write u as a function of x.
2. Take the derivative du/dx, and solve for the quantity dx in terms of du.
3. Use the expression you obtain in part 2 to substitute for dx in the given integral.

Deciding What to Use for u

There is no hard and fast rule, but some guidelines that sometimes work are the following.

• Take u to be an expression that is being raised to a power.
• Take u to be an expression whose derivative appears as a factor of the integrand.
• Take u to be the denominator is a rational expression.
• If the variable x cannot be eliminated using a substitution, try another substitution.
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Example

To evaluate (x2+1)(x3 + 3x - 2)2 dx, proceed as follows.

u = x3 + 3x - 2     Decide on a choice of u
du

dx
= 3x2 + 3
=3(x2 + 1)     Take du/dx
dx =
 du3(x2 + 1)
Solve for dx

Now substitute in the integral to obtain the solution, as follows:

 (x2+1)(x3 + 3x - 2)2 dx

=  (x2+1)u2 du3(x2 + 1)
Substitute for u and dx
=  u2 du3
Cancel to eliminate x
=  13 u2 du
Constant multiple rule
=  13 u33 + C
Do the integration
=  (x3+3x-2)39 + C
Substitute back for x

Want some practice? Try the tutorial or exercises.

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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

v(t) = a(t) dt and

s(t) = v(t) dt.

Moreover, for motion due to gravity close to the earth's surface, ignoring air resistance, a(t) ≈ -32 ft/s2 is constant. Integrating this twice gives the equations

v(t) = v0 - 32t ft/sec     and

s(t) = s0 + v0t -16t2

ft

where v0 is the initial velocity and s0 is the initial position.

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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:

Δx = (b-a)/n,
x0 = a,
x1 = a + Δx,
x2 = a + 2Δx,
...
xn = a + nΔx = b

Next, add up the n products f(x0)Δx, f(x1)Δx, f(x2)Δx, ..., f(xn -1)Δx, to get the Riemann sum.

Thus,

(Left) Riemann sum =
 n-1n = 0 f(xk)Δx
=
 f(x0)Δx + f(x1)Δx + ... + f(xn -1)Δx
=
 [f(x0) + f(x1) + ... + f(xn -1)]Δx
The left Riemann sum gives the area shown below.

The Definite Integral

If f is a continuous function, the definite integral of f from a to b is defined as

 b  a f(x) dx = limn n-1n = 0 f(xk)Δx

In Words:

The definite integral sum is the limit of the Riemann sums as the number of subdivisions gets larger and larger.

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.

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Example

Let us compute the Riemann sum for the integral-11(x2+1) dx using n = 5 subdivisions.

First, to compute the subdivisions:

Δx = (b-a)/n = (1-(-1)/4 = 0.4.
x0 = a = -1
x1 = a + Δx = -1 + 0.4 = 0.6
x2 = a + 2Δx = -1 + 2(0.4) = 0.2
x3 = a + 3Δx = -1 + 3(0.4) = 0.2
x4 = a + 4Δx = -1 + 4(0.4) = 0.6
x5 = b = 1

The Riemann sum we want is

f(x0x + f(x1x + ... + f(x4x
= [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.

 x -1 -0.6 -0.2 0.2 0.6 Total f(x) = x2+1 2 1.36 1.04 1.04 1.36 6.8

The Riemann sum is therefore

6.8Δx = 6.80.4 = 2.72.

To obtain an approximation of the integral, we need to use a much larger number of subdivisions than 5, and technology is necessary for this. You can compute left Reimann sums on-line now with the Numerical Integration Utility.

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.

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The Definite Integral as Area: Geometric Approach

Geometric Interpretation of the Definite Integral (Non-Negative Functions)

If f(x) ≥ 0 for all x in [a, b], then abf(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, abf(x) dx is the area between x = a and x = b that is above the x-axis and below the graph of f, minus the area that is below the x-axis and above the graph of f.

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Example
1.
 2  0 x dx = 2

Area Shown = 2
2.
 1  -1 x dx = 0

Green - Red = 0

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 01 (1-x2) dx.

If Δx is the width of each rectangle, then:

Area of the first (leftmost) rectangle = height × width = f(0)Δx = f(x0)Δx
Area of the second rectangle = height × width = f(x1)Δx
....
Area of the last rectangle = height × width = f(xn -1)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,

Limit of Riemann Sum = Integral = Area

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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) = ax 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

 b  a f(x) dx = F(b) - F(a).

Part (b) in words:

To compute the definite integral ∫abf(x) dx without having to use Riemann sums, all we need do is find an antiderivative of f, evaluate it at x = b, evaluate it at x = a, and subtract.

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Example

Example of (a)
 If A(x) = x  0 et2 dt,  then  A'(x) = ex2.

Example of (b)
Since F(x) = x2 is an antiderivative of f(x) = 2x,
 1  0 2x dx   =   F(1) - F(0) = 12 - 02 = 1.

Another Example of (b)

 1  -1 (1-x2) dx = x - x3/3 1  -1 = 23 - (- 23 ) = 43

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Last Updated: June 2007