It took a long time to create the riemann integral, that it a mathematically rigourous method to find the anti-derivative or primative of a function. The riemann integral of a curve is the area underneath the curve. %%visits: 5
What is the definition of the Rienmann Integral? The (Riemann)
integral := The Riemann integral of a function f (x) on the
interval [a, b], if it exists, is equal to the sum of the areas of the
steps of any staircase function S(x) in the limit that the step width,
w, approaches zero so long as limw → 0 [S(x)] = f (x).
# diary :I’ve realied how bad I am at understanding the simple things,
as I should be able to rewrite my whole wiki on the fundamental topics
that are here. I see that my system has not been optimised. # intuition
- we consider two staircase functions (which is informally just a
function that looks like stairs), namely, the S− and S+. I think of these as
an overestimation and an underestimation of the function, and what we
do, is that we say as the approximations get closer and closer, they get
closer to the real value of the integral. As when $^{a}{b}
S^{-}(x)dx ^{a}{b} f(x) ^{a}_{b} S^{+} $ the limits sandwich and
become equal to the integral of f(x) - It is an infinite sum that
therefore approaches a value, we never know with infinity, but
mathematically we say the sum approaches infinty - ∫U∫f(x, y)dxdy
volume under graph f ,in analogy we have $ _U ^3 g(x,y,z) dx dy dz $ as
a 4 dimensional volume under the graph bounded by the space U ⊂ ℝ3 - def: $U
g(x,y,z) dx dy dz $ = $\lim_{M,N,L \to \infty}
\sum_{i,j,k = 0}^{M,N,L} g(x_i,y_j,z_k) \partial x_i \partial y_j
\partial z_k$ - remark: $ {}{U ^{3}} 1 dx dy dz $ is just
the region U - ∬Rf(x, y) dA
= ∬Rf(x, y) dy dx
if the region R permits - ∫ab∫cdf(x, y) dy dx = ∫ab(∫cdf(x, y) dy)dx.
A fun idea for [[unbouned_integral]]s is to think of them as [[improper_integral]]s but slipped on the x = y axis
We say that a function f is Riemann integrable on an interval [a, b], (a < b) and that it’s Riemann integrable over [a, b] equals ℐ: ∫abf(x)dx = ℐ,
if for every ϵ > 0 there exists δ > 0 such that the following holds:
for any a = x1 < x2 < x3…xn < xn + 1 = b with $$ \underset{k=1,\ldots,n}{max}(x_{k+1} -x_k) \le \delta $$
and any ξ ∈ [xk, xk + 1], one has $$ \left| \mathcal{I} - \sum _{k=1}^{n} f(\xi)(x_{k+1}-x_k) \right| \le \epsilon $$
The (Riemann) integral := he Riemann integral of a
function f (x) on the interval [a, b], if it exists, is equal to the sum
of the areas of the steps of any staircase function S(x) in the limit
that the step width, w, approaches zero so long as limw → 0 [S(x)] = f (x).
$_U g(x,y,z) dx dy dz $ = $\lim_{M,N,L \to
\infty} \sum_{i,j,k = 0}^{M,N,L} g(x_i,y_j,z_k) \partial x_i \partial
y_j \partial z_k$ # exam clinic Think, integrate the awkward line
first. # examples tags :math:
:todo:todo:calculus_2:calculus_1:real_analysis:
{{file:../figures/screenshot_20211123_093755.png}}
{{file:../figures/screenshot_20211123_100123.png}}
{{file:../figures/screenshot_20211123_102245.png}}
{{file:../figures/screenshot_20211216_144855.png}}