Introduction to Integration and Riemann Sums

Integration and the Area Under a Curve

Introduction to Integration

Integration is a fundamental concept in calculus, often described as the opposite of differentiation. It is primarily used to find the area under a curve, which represents the accumulation of quantities such as distance, area, or probability.

• Integration is the process of finding the total accumulation of a quantity, often visualized as the area under a curve on a graph.

• It is the inverse operation of differentiation.

• Key application: Estimating the area between a function and the x-axis.

The Area Under the Curve Problem

Defining the Area

The area under a curve refers to the region bounded by the graph of a function and the x-axis, between two specified points a and b.

• For a function f(x), the area under the curve from x = a to x = b is the region of interest.

• This area can be interpreted as the integral of f(x) over the interval [a, b].

Areas of Simple Shapes

Rectangles and Triangles

• For constant functions (e.g., f(x) = 1), the area under the curve is a rectangle: $A = l \times w$

• For linear functions (e.g., f(x) = x), the area under the curve from 0 to 1 forms a right triangle: $A = \frac{1}{2} b h$

• For nonlinear functions (e.g., f(x) = x^2), there is no simple geometric formula for the area under the curve.

Approximating Areas: The Usefulness of Rectangles

Rectangular Approximation

When the area under a curve cannot be found using basic geometric formulas, we approximate it using rectangles. This method is the foundation of Riemann sums.

• Approximation involves dividing the interval [a, b] into n subintervals of equal width.

• The width of each rectangle is $\Delta x = \frac{b-a}{n}$.

• The height of each rectangle is determined by the value of the function at a chosen point within the subinterval (left endpoint, right endpoint, or midpoint).

Riemann Sums

Definition and Construction

A Riemann Sum is a method for approximating the area under a curve by summing the areas of multiple rectangles.

• Divide the interval [a, b] into n subintervals, each of width $\Delta x$.

• For each subinterval, construct a rectangle whose height is determined by the function value at a specific point (left, right, or midpoint).

• The sum of the areas of these rectangles approximates the total area under the curve.

Types of Riemann Sums

• Left Endpoint Approximation: Use the left endpoint of each subinterval to determine the height.

• Right Endpoint Approximation: Use the right endpoint of each subinterval.

• Midpoint Approximation: Use the midpoint of each subinterval.

Formulas for Riemann Sums

• General Riemann Sum: $\sum_{i=1}^{n} f(x_i^*) \Delta x$ where $x_i^*$ is the chosen point in the i-th subinterval.

• For left endpoints: $x_i^* = a + (i-1)\Delta x$

• For right endpoints: $x_i^* = a + i\Delta x$

• For midpoints: $x_i^* = a + (i-0.5)\Delta x$

Examples

• Approximating the area under $f(x) = x^2$ from 0 to 1 using 3 rectangles:

  • Left endpoints: $\frac{1}{27} + \frac{4}{27} = \frac{5}{27} \approx 0.18519$ (underestimation)

  • Midpoints: $\frac{1}{108} + \frac{9}{108} + \frac{25}{108} = \frac{35}{108} \approx 0.32407$

  • Right endpoints: $\frac{4}{27} + \frac{9}{27} + \frac{14}{27} = \frac{14}{27} \approx 0.51852$ (overestimation)

Generalization and the Definite Integral

Limit of Riemann Sums

As the number of rectangles n increases, the approximation becomes more accurate. In the limit as n approaches infinity, the Riemann sum converges to the exact area under the curve, which is the definite integral.

• Definite Integral: $\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$

• As $n \to \infty$, the placement of endpoints becomes irrelevant: $L_n = M_n = R_n$

Summary Table: Endpoint Formulas

Type	Formula for $x_i^*$	Interval for $i$
Left Endpoint	$x_i^* = a + (i-1)\Delta x$	$i = 1, 2, ..., n$
Right Endpoint	$x_i^* = a + i\Delta x$	$i = 1, 2, ..., n$
Midpoint	$x_i^* = a + (i-0.5)\Delta x$	$i = 1, 2, ..., n$

Key Takeaways

• Rectangles can be used to approximate the area under a curve, leading to the concept of the Riemann sum.

• The more rectangles used, the better the approximation.

• In the limit, the Riemann sum becomes the definite integral, giving the exact area under the curve.