Calculate $\Delta x = \frac{b-a}{n}$.

Use the formula $x_i = a + i\Delta x$.

1. Identify a, b, and f(x). 2. Express the limit of the Riemann sum in the form $\int_a^b f(x) dx$.

1. Find $\Delta x$. 2. Find $x_i$. 3. Evaluate $f(x_i)$. 4. Compute $\sum_{i=1}^n f(x_i) \Delta x$.

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

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

1. Calculate $\Delta x$. 2. Find $x_i$ for each subinterval. 3. Evaluate $f(x_i)$ for each $x_i$. 4. Sum the areas: $\sum_{i=1}^{10} f(x_i) \Delta x$.

Use the formula $\Delta x = \frac{b-a}{n}$, where $a$ and $b$ are the limits of integration.

Use the formula $x_i = a + i\Delta x = a + i\frac{b-a}{n}$, where $a$ is the lower limit of integration.

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

$\Delta x = \frac{b-a}{n}$

$x_i = a + \Delta x cdot i$

$\sum_{i=0}^{n-1} {\Delta x}cdot{f({x_i})}$

$\sum_{i=1}^{n} {\Delta x}cdot{f({x_i})}$

$\lim_{n\to \infty}\sum_{i=1}^{n} {\Delta x}cdot{f({x_i})}=\int_a^bf(x) dx$

$A(i) = \Delta x * f(x_i)$

Substitute $x_i$ into the original function $f(x)$.

$\int_a^b f(x) dx$

$\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x = \int_a^b f(x) dx$

$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(a + i\Delta x) \Delta x$, where $\Delta x = \frac{b-a}{n}$

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

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

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

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

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.

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

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

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

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

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.