Written like this this, we see the relationship between the definite integral(left side) and the net change of the antiderivative(right side)
This gives us a way to compute definite integrals. The process is
1. find the antiderivative
2. Evaluate it at the limits of integration
∫ba g′(x)dx=g(b)−g(a).
We will be relating a definite integral to the net change(integral function) in its antiderivative.
it is a running total of the amount of stuff that f represents, between a and x. If f is the height of a curve, then I(x) is the area under the curve between a and x. If f is velocity, then I(x) is the distance traveled between time a and time x.
I(x)=∫xaf(s)ds.
Definite Integral
Integral Function
The Fundamental Theorem
A = This limit + summation + f(xk)Δx
Now we want to combine this condensed summation with the increase in subdivisions. We are going to need infinite subdivisions, and the best way to achieve that is through limits.
We will collapse this into a summation,
Ln = this summation^f(xk)Δx
(Keep in mind RH summation would look similar, but k = 1 instead)
Because we are using the left point as the height, f1 is 0, and Ln is as follows.
Ln = f (x0)Δx + f (x1)Δx + ...f (xn-1)Δx
We will be generalizing the previous left hand method. To do this, we will be finding the area under the graph f between a and b, with n subdivisions.
The problem with these three is that they are not very exact. To fix this, we want to increase our sub-divisions of the graph from three to a much larger number.
Mid-Point approximation
For the third method, we would use the value of the function at the midpoint of each of the subdivisions as the height.
Area as a Sum
Right Hand Approximation
We would do the same staring process for this method. We then use the value of the function at the right endpoint of each subdivision as the height.
Left Hand Approximation
We would divide a curve(for example y=x^2) into 3 even sections along the x-axis. This would be 0-1/3, 1/3-2/3, and 2/3-1. We then add the areas under the curve.
Examples
integral sin(2x)dx
=
-cos(2x)1/2 + c
Integral (3x + 4)dx
=
3(x^2/2) + 4x + C
Integral 4dx
=
4x + C
sin(x)dx = - cos(x) + c
cos(x)dx = sin(x) + c
sec2(x)dx = tan(x) + c
This theorem allows us to compute an antiderivative by treating a function term-by-term and factoring out constants
2: If f is a continuous function and k is a real constant
1: If f and g are continuous functions
Derivative
A Derivative is a
way to show instantaneous rate of change
Basic
function = general antiderivative
x^n = (1/n+1)x^(n+1)+c
(ax+b)^n = 1/a(n+1)(ax+b)^(n+1)+c
Antiderivative
In order to calculate integrals, we now see that it's important to be able to find antiderivatives of functions. An antiderivative of a function f(x) is a function whose derivative is equal to f(x). That is, if F′(x)=f(x), then F(x) is an antiderivative of f(x).
A given function can have many antiderivatives. For instance, the following functions are all antiderivatives of x2 : x^3/3, x^3/3 +1, x^3/3 + 42, and more.
Because any two antiderivatives differ by a constant
c, we can write a general antiderivative. For x^2, it would be x^3/3 + C.
Theorem (Linearity of integration)
1: ∫(f(x)±g(x))dx = ∫f(x)dx±∫g(x)dx
2: ∫ kf(x)dx = k∫f(x)dx
Integral
Indefinite Integral
An indefinite integral is an integral written without terminals; it simply asks us to find a general antiderivative of the integrand. It is not one function but a family of functions
∫ x^2 dx= x^3/3 + c
