Area, Finite Sums, and the Definite Integral

Section 5.1: Area and Estimating with Finite Sums

Basic Area Formulas

Understanding the area of basic geometric shapes is foundational for calculus, as it leads to the concept of integration. The area of more complex regions can often be approximated by decomposing them into simpler shapes.

• Rectangle: The area is the product of its length and width.

• Triangle: The area is half the product of its base and height.

• Polygon: The area can be found by dividing the polygon into triangles and summing their areas.

Estimating Area Under Curves

For regions bounded by curves, such as the area under y = 1 - x^2, there is no simple geometric formula. Instead, we approximate the area using rectangles, a process that leads to the concept of the definite integral.

• Upper and Lower Estimates: By inscribing or circumscribing rectangles under or over the curve, we obtain lower and upper bounds for the area.

• Increasing Accuracy: Using more rectangles (subintervals) improves the approximation.

Finite Approximations and Sums

As the number of rectangles increases, the sum of their areas approaches the true area under the curve. This process is formalized using finite sums and, ultimately, limits.

• Lower, Midpoint, and Upper Sums: These are different methods for choosing the height of each rectangle (left endpoint, midpoint, right endpoint).

• Tabular Comparison: The following table shows how the approximation improves as the number of subintervals increases.

Number of subintervals	Lower sum	Midpoint sum	Upper sum
2	0.375	0.6875	0.875
4	0.53125	0.617875	0.78125
16	0.634765625	0.6669921875	0.697265625
50	0.6566	0.6667	0.6766
100	0.66165	0.666675	0.67165
1000	0.6661665	0.66666675	0.6671665

Section 5.2: Sigma Notation and Limits of Finite Sums

Sigma Notation

Sigma notation provides a concise way to write sums, especially those with many terms. It is essential for expressing Riemann sums and the process of integration.

• Summation Symbol: $\sum_{k=1}^n a_k$ represents the sum of terms $a_k$ from $k=1$ to $n$.

• Examples: Common sums include arithmetic and geometric series, as well as sums involving powers of integers.

Algebraic Rules for Sums

Finite sums obey several algebraic rules, which are useful for manipulating and simplifying expressions before taking limits.

• Sum Rule: $\sum_{k=1}^n (a_k + b_k) = \sum_{k=1}^n a_k + \sum_{k=1}^n b_k$

• Difference Rule: $\sum_{k=1}^n (a_k - b_k) = \sum_{k=1}^n a_k - \sum_{k=1}^n b_k$

• Constant Multiple Rule: $\sum_{k=1}^n c a_k = c \sum_{k=1}^n a_k$

• Constant Value Rule: $\sum_{k=1}^n c = n c$

Special Sums

Some sums have closed-form formulas, which are especially useful in evaluating Riemann sums.

• $\sum_{k=1}^n k = \frac{n(n+1)}{2}$

• $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$

• $\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2$

Section 5.3: The Definite Integral

Definition and Interpretation

The definite integral is the limit of Riemann sums as the number of subintervals approaches infinity. It represents the signed area under a curve between two points.

• Riemann Sum: $\sum_{k=1}^n f(c_k) \Delta x_k$ where $c_k$ is a sample point in the $k$th subinterval and $\Delta x_k$ is the width of the subinterval.

• Definite Integral: $\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{k=1}^n f(c_k) \Delta x_k$

• Geometric Meaning: The definite integral gives the net area between the graph of $f(x)$ and the $x$-axis from $x=a$ to $x=b$.

Properties and Theorems

The definite integral has several important properties that facilitate its computation and application.

• Order of Integration: $\int_a^b f(x) dx = -\int_b^a f(x) dx$

• Zero Width Interval: $\int_a^a f(x) dx = 0$

• Linearity: $\int_a^b [cf(x) + g(x)] dx = c\int_a^b f(x) dx + \int_a^b g(x) dx$

• Additivity: $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ for $a < c < b$

• Integrability: If $f$ is continuous on $[a, b]$, or has finitely many jump discontinuities, then $f$ is integrable on $[a, b]$.

Applications and Examples

Definite integrals are used to compute areas, average values, and other quantities. For example, the area under $y = x^2$ from $x=0$ to $x=1$ can be approximated by Riemann sums and exactly computed as $\int_0^1 x^2 dx = \frac{1}{3}$.

• Example: Approximating the area under $y = x^2$ using right endpoints and $n$ rectangles leads to the sum $R_n = \frac{1}{n^3} \sum_{i=1}^n i^2$.

• Limit: $\lim_{n \to \infty} R_n = \frac{1}{3}$

Summary Table: Rules Satisfied by Definite Integrals

Rule	Statement
Order of Integration	$\int_a^b f(x) dx = -\int_b^a f(x) dx$
Zero Width Interval	$\int_a^a f(x) dx = 0$
Constant Multiple	$\int_a^b k f(x) dx = k \int_a^b f(x) dx$
Sum and Difference	$\int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx$
Additivity	$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$
Max-Min Inequality	If $m \leq f(x) \leq M$ on $[a, b]$, then $m(b-a) \leq \int_a^b f(x) dx \leq M(b-a)$
Domination	If $f(x) \geq g(x)$ on $[a, b]$, then $\int_a^b f(x) dx \geq \int_a^b g(x) dx$

Additional info: These notes cover the foundational concepts of area, finite sums, sigma notation, and the definition and properties of the definite integral, which are central to calculus chapters on integration.