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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.

Why use Riemann Sums?

To approximate the area under a curve, especially when an exact formula is not available.

How does increasing 'n' affect Riemann Sums?

Increasing 'n' (number of subintervals) increases the accuracy of the approximation.

What is the relationship between Riemann Sums and Definite Integrals?

The definite integral is the limit of the Riemann sum as the number of subintervals approaches infinity.

Explain the concept of summation notation.

Summation notation provides a compact way to represent the sum of a series of terms, making calculations and manipulations more efficient.

Explain the connection between Riemann sums and definite integrals.

Definite integrals represent the exact area under a curve, which can be approximated by Riemann sums. As the number of subintervals approaches infinity, the Riemann sum converges to the definite integral.

Why is the limit needed when defining the definite integral using Riemann sums?

The limit ensures that the approximation becomes exact as the width of the rectangles approaches zero and the number of rectangles approaches infinity.

Describe how the choice of left or right endpoints affects the Riemann sum approximation.

Left endpoints may underestimate the area if the function is increasing, while right endpoints may overestimate. The opposite is true for decreasing functions.

What does the definite integral represent?

The definite integral represents the signed area between the curve of a function and the x-axis over a specified interval.

How does the width of the subintervals affect the accuracy of the Riemann sum?

Smaller subintervals generally lead to a more accurate approximation because they reduce the error between the rectangles and the curve.

What is the significance of expressing a definite integral as the limit of a Riemann sum?

It provides a rigorous definition of the definite integral and connects it to the concept of area under a curve, allowing for calculations and analysis of complex functions.

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.

Why use Riemann Sums?

To approximate the area under a curve, especially when an exact formula is not available.

How does increasing 'n' affect Riemann Sums?

Increasing 'n' (number of subintervals) increases the accuracy of the approximation.

What is the relationship between Riemann Sums and Definite Integrals?

The definite integral is the limit of the Riemann sum as the number of subintervals approaches infinity.

Explain the concept of summation notation.

Summation notation provides a compact way to represent the sum of a series of terms, making calculations and manipulations more efficient.

Explain the connection between Riemann sums and definite integrals.

Why is the limit needed when defining the definite integral using Riemann sums?

The limit ensures that the approximation becomes exact as the width of the rectangles approaches zero and the number of rectangles approaches infinity.

Describe how the choice of left or right endpoints affects the Riemann sum approximation.

Left endpoints may underestimate the area if the function is increasing, while right endpoints may overestimate. The opposite is true for decreasing functions.

What does the definite integral represent?

The definite integral represents the signed area between the curve of a function and the x-axis over a specified interval.

How does the width of the subintervals affect the accuracy of the Riemann sum?

Smaller subintervals generally lead to a more accurate approximation because they reduce the error between the rectangles and the curve.

What is the significance of expressing a definite integral as the limit of a Riemann sum?

It provides a rigorous definition of the definite integral and connects it to the concept of area under a curve, allowing for calculations and analysis of complex functions.

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