Home/Calculus II/Definite integral and Fundamental Theorem of Calculus/Riemann sumsSection 3.1Riemann sumsFor a partition a=x0<x1<⋯<xn=ba=x_0<x_1<\cdots<x_n=ba=x0<x1<⋯<xn=b, with Δxi=xi−xi−1\Delta x_i=x_i-x_{i-1}Δxi=xi−xi−1 and sample points xi∗x_i^*xi∗,FormulavizCopy LaTeX∫abf(x) dx=lim∥P∥→0∑i=1nf(xi∗)Δxi,\int_a^b f(x)\,dx =\lim_{\|P\|\to0}\sum_{i=1}^{n}f(x_i^*)\Delta x_i,∫abf(x)dx=∥P∥→0limi=1∑nf(xi∗)Δxi,View detail →if the limit exists.For a uniform partition, Δx=(b−a)/n\Delta x=(b-a)/nΔx=(b−a)/n:Copy LaTeX∫abf(x) dx=limn→∞∑i=1nf(xi∗)Δx.\int_a^b f(x)\,dx =\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\Delta x.∫abf(x)dx=n→∞limi=1∑nf(xi∗)Δx.View detail →
FormulavizCopy LaTeX∫abf(x) dx=lim∥P∥→0∑i=1nf(xi∗)Δxi,\int_a^b f(x)\,dx =\lim_{\|P\|\to0}\sum_{i=1}^{n}f(x_i^*)\Delta x_i,∫abf(x)dx=∥P∥→0limi=1∑nf(xi∗)Δxi,View detail →
For a uniform partition, Δx=(b−a)/n\Delta x=(b-a)/nΔx=(b−a)/n:Copy LaTeX∫abf(x) dx=limn→∞∑i=1nf(xi∗)Δx.\int_a^b f(x)\,dx =\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\Delta x.∫abf(x)dx=n→∞limi=1∑nf(xi∗)Δx.View detail →