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Define Riemann Sum.

Approximation of the area under a curve by dividing it into rectangles.

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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.

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?

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?

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?

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?

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?

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?

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: $\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x$ .

All Flashcards

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.

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?

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?

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?

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?

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?

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?

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: $\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x$ .