Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 541

• We presented just a few pen-and-paper rules for differentiation. Dif-

ferentiation is a linear operator, which allows us to differentiate sums.
The power rule tells us how to evaluate expressions of the form

dt t n .
Together, these rules allow us to take derivatives of polynomials. We
also presented the derivatives for the sine, cosine, and exponential
function. The chain rule tells us how to differentiate a function of the
form f(g(t)).

d

• An integral is a “continuous summation,” or “running total.” These

sums are also equivalent to the area under the graph of the function
being summed.

• A Riemann integral defines an integral using a limit argument. We

take the sum of a large number of small elements, which in general
is an approximation to the true sum when the number of elements is
finite. The true sum is obtained by considering what happens as we
increase the number of elements to infinity, causing the error in our
approximation to vanish.

• Riemann integrals are usually not directly solvable in the same way

that derivatives are. They are used to recognize when the problem
we are solving is an integral, and to help set up the integral properly.
It's also how we solve them numerically (we have not yet discussed
the details of how to do this).

• The fundamental theorem of calculus says that integration and dif-

ferentiation are inverse operations. On paper, definite integrals are
computed by looking for an antiderivative, not by evaluating the Rie-
mann integral at the limit. A function whose argument defines the
upper limit of integration will be an antiderivative of the integrand.

• An indefinite integral is a function that is an antiderivative of the

integrand. A definite integral produces a number representing the
continuous summation of the integrand over the interval identified
by the limits of integration. A definite integral can be calculated by
evaluating any antiderivative at the starting and ending points, and
taking the difference between these two values (by subtracting the
value at the start of the interval from the value at the end of the
interval). An indefinite integral is actually a definite integral where
the limits of integration are implied.