These are my complete notes for Definite Integrals in Integral Calculus.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@berkeley.edu.
Summary of Definite Integrals (Integral Calculus)
?. Definite Integrals.
# Integral Symbol:
$$\int_a^b f(x)$$ a = lower limit (with respect to the axis of the variable of integration)
b = upper limit (with respect to the axis of the variable of integration)
f(x) = Integrand, function
dx = Variable of Integration
# Definite Integral: If y = f(x) is nonnegative & integrable over a closed interval [a, b], then the area under the curve y = f(x) from a to b is the integral of f from a to b.
$$A = \int_a^b f(x)\,dx$$ 'a' and 'b' are the lower and upper x-bounds of the curve, the x-ness of which is given by the 'dx' at the end of the equation.
This definition works both ways. Integrals can be used to calculate Area AND you can find area to calculate integrals.
#
Rule .
Rules for Definite Integrals:
- $$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$$
The NEGATIVE Rule. - $$\int_a^a f(x)\,dx = 0$$
The LINE Rule. - $$\int_a^b cf(x)\,dx = c\int_a^b f(x)\,dx$$
Constant Multiple Rule. - $$\int_a^b f(x)±g(x)\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx$$
The SUM Rule. - $$\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$$
The RECOMBINATION Rule. - The Max/Min rule is just like the squeeze theorem.
#
Rule .
The derivative of an integral is the original function. The integral is the antiderivative of f. If F is an antiderivative of f, then:
$$\int_a^x f(t)\,dt = F(x) + c$$
where c is some constant. By setting x = a,
$$\int_a^x f(t)\,dt = F(a) + c = 0$$
$$-F(a) = c$$
