Which we will use to evaluate definite integrals. The first part of the fundamental theorem of calculus also provides a powerful corollary, This constant or set it equal to zero since it is cancelled out, as we shall see. However, for definite integrals, we can ignore This is to elucidate the fact that there can be infinitely many antiderivatives Term, which we usually add after integrating 𝑓 ( 𝑥 ),īut we have explicitly stated it here to examine the indefinite integral. We note that the constant of integration 𝐶 is included in the first When 𝑓 is continuous and there are infinitely many antiderivativesįor 𝑓, obtained by adding this arbitrary constant to 𝐹. By the first part of the theorem, antiderivatives of 𝑓 always exist Where 𝐶 is known as the constant of integration. 𝑓 by computing the indefinite integral of 𝑓 given as In other words, we can compute the antiderivative 𝐹 of some function If 𝑓 is a continuous real-valued function defined onįunction defined, for all 𝑥 in, as The First Part of the Fundamental Theorem of Calculus While below the 𝑥-axis it is always negative, as shown in the diagram. The area given by the definite integral above the 𝑥-axis is always positive, The definite integral always gives the signed area under the curve The function 𝑓 is evaluated at the midpoint of each subinterval, 𝑓 is evaluated at the left endpoint of each subinterval, If 𝑥 = 𝑥 ∗    , that is, the function.Or definite integral, for simplicity, and it corresponds to the example above withĪn estimate of the area under the curve using 𝑛 equal-width rectangles and the diagram, This is the choice that most people use when finding a specific Riemann sum The definite integral in terms of this sum is 𝑓 is evaluated at the right endpoint of each subinterval, If 𝑥 = 𝑥 ∗  , that is, the function.In particular, the common choices are given by the following: 𝑥 ∗  is arbitrary, which may produce different Riemann sums, Or width of the summands Δ 𝑥 → 0, so does the difference betweenĪny two points in the interval. It does not matter which sample point 𝑥 ∗  in the subinterval Provided that the limit exists and gives the same value for all sample points The definite integral from 𝑥 = 𝑎 to 𝑥 = 𝑏 is defined in     of equal width, Δ 𝑥,Īnd choose sample points 𝑥 ∈ ∗     . Interval, we can divide the interval into 𝑛 subintervals Given a function 𝑓 that is continuous and defined on the

Fundamental of calculus part 1