Calculus I: Area Problem and Definite Integrals

Definite Integrals

• Definite integrals are the second kind of integral, focused on the area problem.

Area Problem

• The area problem aims to find the area between a positive function and the x-axis over an interval.
• Example: For function $f(x) = x^2 + 1$ on interval $[0, 2]$.

Estimation Techniques

• Creating Rectangles:
  • Divide the interval into $n$ subintervals of width $\frac{b-a}{n}$.
  • Estimate area using rectangles based on the function's height at specific points (right, left, or midpoint).

Types of Endpoint Heights:

1. Right Endpoint:

   • Overestimates area (each rectangle adds more area than below the curve).
   • Formula: $A_r = ext{Height} imes ext{Width}$

2. Left Endpoint:

   • Underestimates area (each rectangle misses some area).

3. Midpoint:

   • Provides a better approximation than endpoints.

Comparing Estimates

• Exact area calculated for functions results in:
• Right Endpoint: $A_r = 5.75$,
• Left Endpoint: $A_l = 4.18$,
• Midpoint: $A_m = 4.66$.

Increasing Rectangles

• More rectangles lead to better approximations.
• Using midpoint heights often provides the best estimates.

General Case

• For any function $f(x)$, divided into $n$ subintervals:
• Notation: $A ext{ approximated as } A = \frac{b-a}{n} imes (f(x<em>1) + f(x</em>2) + … + f(x_n))$
• This is a Riemann Sum.
• For better approximations, let $n o ext{infinity}$.

Negative Functions

• If functions are below the x-axis, estimate using the same method gives negative area.
• Functions above axis yield positive area; mix results yield 'net area'.
• Examples include functions defined below and above the x-axis leading to calculating a net area.

Definite Integrals - Definite integrals are the second kind of integral, focused on the area problem. ### Area Problem - The area problem aims to find the area between a positive function and the x-axis over an interval. - Example: For function $f(x) = x^2 + 1$ on interval $[0, 2]$. ### Estimation Techniques - **Creating Rectangles**:- Divide the interval into $n$ subintervals of width $\frac{b-a}{n}$. - Estimate area using rectangles based on the function's height at specific points (right, left, or midpoint). #### Types of Endpoint Heights: 1. **Right Endpoint**:- Overestimates area (each rectangle adds more area than below the curve). - Formula: $A_r = \text{Height} \times \text{Width}$ 2. **Left Endpoint**:- Underestimates area (each rectangle misses some area). 3. **Midpoint**:- Provides a better approximation than endpoints. ### Comparing Estimates - Exact area calculated for functions results in: - Right Endpoint: $A_r = 5.75$, - Left Endpoint: $A_l = 4.18$, - Midpoint: $A_m = 4.66$. ### Increasing Rectangles - More rectangles lead to better approximations. - Using midpoint heights often provides the best estimates. ### General Case - For any function $f(x)$, divided into $n$ subintervals: - Notation: $A \text{ approximated as } A = \frac{b-a}{n} \times (f(x_1) + f(x_2) + \ldots + f(x_n))$. - This is a **Riemann Sum**. - For better approximations, let $n \to \text{infinity}$. ### Negative Functions - If functions are below the x-axis, estimate using the same method gives negative area. - Functions above axis yield positive area; mix results yield 'net area'. - Examples include functions defined below and above the x-axis leading to calculating a net area. ### Practice Problems 1. Estimate the area under the curve $f(x) = x^3$ from $x = 1$ to $x = 3$ using right endpoints with $n = 4$ rectangles. 2. Estimate the area under the curve $f(x) = \sin(x)$ from $x = 0$ to $x = \pi$ using midpoint rule with $n = 3$ rectangles. 3. Determine the net area of the function $f(x) = x^2 - 4$ from $x = -2$ to $x = 3$ using definite integrals. 4. Approximate the area under the curve $f(x) = e^x$ from $x = 0$ to $x = 2$ using left endpoints with $n = 5$ rectangles.