#The Fundamental thereom of calculus The fundamental thereom of calculus allows you to calculate the area under the curve exactly. The fundamental thereom of calculus takes advtange of antiderivatives. While the fundamental thereom of calculus may not be applicable in every situation, it should be your preferred choice as its accuracey is as good as your calculator, and it usually takes less work then using sums or one of the formulas for numerical integration that we will look at in the next section. Now, lets get to it. ##Definite Integrals While in the previous two note sets we have been looking at the area \(A\) under the curve in a closed interval \([a,b]\) there is a specific notation for this. It is quite similar to the definite integral. It looks like this: \[ \int_{a}^{b}f(x)dx=A \] While the notation may be a little bit weird at first, especially since \(b\) is on top and \(a\) is in the bottom, but after a while it will make quite a bit of sense. The \(dx\) after the function has the same use as it does in indefinite integrals. It tells you which variable to integrate on. If we rewrite the definition of a Riemann sum using our fancy new definite integral notation, it looks like this: \[ \int_{a}^{b}f(x)dx=A=\lim_{||p||\to0}\sum_{k=1}^{n}f(w_k)\Delta x_k \] We can also rewrite our original sums, but this would be cumbersome so the formulas have been ommitted. Now, lets look at the fundamental thereom of calculus. ##The Fundamental thereom of calculus The fundamental thereom of calculus allows you to calculate the definite integral exactly rather than achieving a very close approximation with the sums that we have seen in the previous lessons. However, the fundamental thereom of calculus cannot be used everywhere so sometimes we will have to use methods of numerical integration which we will look at in the next note set. Now, for the fundamental thereom of calculus. It can be stated as follows: \[ \int_{a}^{b}f(x)dx = F(b) - F(a) \] where \(F\) is an antiderivative of \(f\). This thereom is able to find the area \(A\) to the degree of accuracey that your calculator can. While this may seem very appealing at first and it is, you may not always be able to find the antiderivative of a function \(f\). In these cases you will want to use numerical integration methods. However, the fundamental thereom of calculus should be your preferred choice.