# Questions tagged [riemann-sum]

This tag is for questions about Riemann sums and Darboux sums.

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### Question about Riemann integrability: do we need to specify that all Riemann sums converge to the same number in the definition?

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Suppose that there is a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$, ...
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### Riemann Sum Approximations: When are trapezoids more accurate than the middle sum?

We can approximate a definite integral, $\int_a^b f(x)dx$, using a variety of Riemann sums. If $T_n$ and $M_n$ are the nth sums using the trapezoid and midpoint (middle) sum methods and if the second ...
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### Trouble integrating 1/x from Riemann Sum

Preface: I'm a A-Level student, so much of the maths I'm speaking about here is quite new to me, in particular Riemann Sums. I apologise if this already has an answer, I couldn't find it. I'm trying ...
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### proof that there is $c \in [a,b]$ such that $f(c) = g(c)$

Let $f,g: [a,b] \rightarrow \mathbb{R}$ continuous functions such that $\int_a^{b} f(x)dx = \int_a^{b}g(x)dx$. Proof that there is $c \in [a,b]$ such that $f(c)=g(c).$ This questions has been asked ...
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### Evaluate the Riemann sum

If $\mathrm{f}\left(x\right) = 2\cos\left(x\right)$ $0 \leq x \leq 3\pi/4$ evaluate the Riemann sum with $n = 6$, taking the sample points to be left endpoints. ( Round your answer to six ...
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### Approximate the area under the curve $f(x) = x^2+4x+6$ on the interval $[2,6]$ using the right-hand Riemann sum

Approximate the area under the curve $f(x) = x^2+4x+6$ on the interval $[2,6]$ using the right-hand Riemann sum where $P$ is the partition of $[2,6]$ determined by $\{2,4,5,6\}$ I set up the right ...
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### How to integrate $xe^x$ without using antiderivatives or integration by parts.

Yesterday, I sat for my Real Analysis II paper. There I found a question asking to integrate $\displaystyle\int_0^1 xe^x \, dx$ without using antiderivatives and integrating by parts. I tried it by ...
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### Counter-Example for Darboux Sums: “Finer” Partition with Greater Difference.

Let $P = \{p_i\}_{i= 1, n}$, $P' = \{p'_j\}_{j = 1, m}$ be partitions of an interval with max$|p'_j| \le$min$|p_i|$, i.e. all the sub-intervals of $P'$ are at least as short as all the sub-intervals ...
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### Arc length of a Polar curve as a Riemann sum

Suppose we have a curve in polar plane satisfying the equation $r=f(\theta)$ with $\theta\in[a,b].$ To find the area enclosed by this curve in this range of $\theta$ using Riemann integrals, we ...
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### Understanding what ij mean in a Double Riemann Sum (Double Integral)

I am having trouble understanding what the ($x^*_{ij}$, $y^*_{ij}$) in this diagram (circled in blue) is explaining. What I do know is that $i$ is the iteration of the $x$ Riemann Sum and the $j$ is ...
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### Cosine of a Wiener process

Let $W_t$ be a standard Brownian motion, i.e., $W_t \sim N(0,t)$. Define the random variable $$X=\int_0^1\cos(W_t)dt$$ A similiar process, $Y_t=\cos(\omega t+\sigma W_t+\theta)$, with the uniform ...
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### What are the requirements of a function so that the left Riemann sum equals the right Riemann sum?

My homework question in particular specifies over an interval of [0,1], the function is negative, and the function is decreasing.
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### Darboux sums elementary question - am I correct

I'm very new to this material and I would like someone more experienced to give input if possible. $Q \subset \mathbb R^n$ is a box and $f: Q \to \mathbb R$ is a function. Let $\Xi_Q$ be the set of ...
### Evaluate integral $\int_{-2}^0 x^2+x\ dx$ using Riemann Sum
Consider the integral $$\int_{-2}^0 x^2+x\ dx.$$ The question says to use Riemann Sum theorem which is $$\sum_{i=1}^nf(x_i)\delta x$$ I know that $\delta x= \frac{-2}{n}$ and that \$x_i=-2+(\frac{2}{n}...