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Formula for $\Delta x$?
$\Delta x = \frac{b-a}{n}$
Formula for $x_i$ (right endpoint)?
$x_i = a + \Delta x cdot i$
General form of Left Riemann Sum?
$\sum_{i=0}^{n-1} {\Delta x}cdot{f({x_i})}$
General form of Right Riemann Sum?
$\sum_{i=1}^{n} {\Delta x}cdot{f({x_i})}$
Definite Integral as Limit of Riemann Sum?
$\lim_{n\to \infty}\sum_{i=1}^{n} {\Delta x}cdot{f({x_i})}=\int_a^bf(x) dx$
Area of i-th rectangle A(i) in Riemann sum?
$A(i) = \Delta x * f(x_i)$
Formula for $f(x_i)$?
Substitute $x_i$ into the original function $f(x)$.
What is the formula for the area under the curve of f(x)?
$\int_a^b f(x) dx$
How to express a limit of a Riemann sum as a definite integral?
$\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x = \int_a^b f(x) dx$
What is the formula for expressing the definite integral as the limit of a Riemann sum?
$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(a + i\Delta x) \Delta x$, where $\Delta x = \frac{b-a}{n}$
How to find $\Delta x$ given interval [a,b] and n?
Calculate $\Delta x = \frac{b-a}{n}$.
How to find $x_i$ for right Riemann sum?
Use the formula $x_i = a + i\Delta x$.
Steps to convert Riemann sum to definite integral?
1. Identify a, b, and f(x). 2. Express the limit of the Riemann sum in the form $\int_a^b f(x) dx$.
How do you evaluate a Riemann sum with given function, interval, and n?
1. Find $\Delta x$. 2. Find $x_i$. 3. Evaluate $f(x_i)$. 4. Compute $\sum_{i=1}^n f(x_i) \Delta x$.
How to express a definite integral as the limit of a Riemann sum?
1. Find $\Delta x = \frac{b-a}{n}$. 2. Find $x_i = a + i\Delta x$. 3. Substitute $x_i$ into $f(x)$. 4. Write the limit: $\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x$.
How to find the summation notation for a right Riemann sum?
1. Determine $\Delta x = \frac{b-a}{n}$. 2. Find $x_i = a + i\Delta x$. 3. Evaluate $f(x_i)$. 4. Write the sum: $\sum_{i=1}^n f(x_i) \Delta x$.
How to calculate the value of a Riemann sum with 10 subintervals?
1. Calculate \(\Delta x\). 2. Find \(x_i\) for each subinterval. 3. Evaluate \(f(x_i)\) for each \(x_i\). 4. Sum the areas: \(\sum_{i=1}^{10} f(x_i) \Delta x\).
How to define \(\Delta x\) in terms of \(n\) when expressing a definite integral as a Riemann sum?
Use the formula \(\Delta x = \frac{b-a}{n}\), where \(a\) and \(b\) are the limits of integration.
How to define \(x_i\) in terms of \(n\) for a right Riemann sum?
Use the formula \(x_i = a + i\Delta x = a + i\frac{b-a}{n}\), where \(a\) is the lower limit of integration.
How to express the definite integral as the limit of a right Riemann sum?
1. Find \(\Delta x = \frac{b-a}{n}\). 2. Find \(x_i = a + i\Delta x\). 3. Substitute \(x_i\) into \(f(x)\). 4. Write the limit: \(\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x\).
Define Riemann Sum.
Approximation of the area under a curve by dividing it into rectangles.
What is Summation Notation?
A concise way to represent the sum of a sequence of numbers.
Define Definite Integral.
The exact area under a curve between two specified limits.
What is $\Delta x$ in Riemann Sums?
The width of each subinterval in the Riemann Sum approximation.
What is $x_i$ in Riemann Sums?
The x-value used to determine the height of the rectangle in the Riemann Sum.
Define Left Riemann Sum.
A Riemann sum where the height of each rectangle is determined by the function value at the left endpoint of the subinterval.
Define Right Riemann Sum.
A Riemann sum where the height of each rectangle is determined by the function value at the right endpoint of the subinterval.
What is the role of $n$ in Riemann Sums?
Represents the number of subintervals used in the approximation.
What does $a$ represent in a definite integral $\int_a^b f(x) dx$?
The lower limit of integration.
What does $b$ represent in a definite integral $\int_a^b f(x) dx$?
The upper limit of integration.