Area and Definite Integrals: Riemann Sums and Applications

7.3 – Area and Definite Integrals

Introduction to Area Under a Curve

In calculus, finding the area under a curve is a fundamental concept, especially when the curve represents a function that is not easily described by simple geometric shapes. This section introduces the idea of using definite integrals to compute such areas, which has direct applications in business, economics, and the life sciences.

• Area under a curve can represent physical quantities such as distance traveled, total revenue, or accumulated change.

• For a velocity function $v(t)$, the area under $v(t)$ from $t=a$ to $t=b$ gives the distance traveled.

• Key formula: $\text{Distance Traveled} = \int_a^b v(t)\,dt$

• When $v(t)$ is constant, the area is simply $v \times (b-a)$.

Example: If a car travels at a constant velocity of 5 m/s for 8 seconds, the distance is $5 \times 8 = 40$ meters.

Estimating Area: Riemann Sums

When the function is not constant or not a simple shape, we estimate the area under the curve by dividing the interval into subintervals and summing the areas of rectangles. This process is called forming a Riemann Sum.

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

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

• Sum the areas of all rectangles to approximate the total area under the curve.

Formula for Riemann Sum:

• Left endpoints: $\sum_{i=0}^{n-1} f(x_i) \Delta x$

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

• Midpoints: $\sum_{i=1}^{n} f(\text{midpoint of } i\text{th interval}) \Delta x$

Example 1: Area Using Left, Right, and Midpoint Endpoints

Consider the region bounded by $y = x^2$ over $[0, 2]$.

• a. Left endpoints: Use the function value at the left of each subinterval.

• b. Right endpoints: Use the function value at the right of each subinterval.

• c. Midpoints: Use the function value at the midpoint of each subinterval.

• d. Increasing number of rectangles: As the number of rectangles increases, the approximation becomes more accurate.

Observation: The midpoint method often gives a better approximation than left or right endpoints for the same number of rectangles.

Example 2: Area Under a Linear Function

Approximate the area under $f(x) = 2x$ over $[0, 4]$ using 4 rectangles of equal width, with heights determined by the midpoint of each interval.

• Calculate $\Delta x = 1$.

• Midpoints: $x = 0.5, 1.5, 2.5, 3.5$.

• Sum: $f(0.5) + f(1.5) + f(2.5) + f(3.5)$, each multiplied by $\Delta x$.

Compare: The actual area under $f(x) = 2x$ from $0$to $4$ is $\int_0^4 2x\,dx = [x^2]_0^4 = 16$.

The Definite Integral

The definite integral is the exact area under the curve from $x=a$ to $x=b$, defined as the limit of the Riemann sum as the number of rectangles approaches infinity.

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

• The definite integral can be positive or negative, depending on whether the function is above or below the $x$-axis.

Key Properties:

• Linearity: $\int_a^b [af(x) + bg(x)]\,dx = a\int_a^b f(x)\,dx + b\int_a^b g(x)\,dx$

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

Example 3: Right Endpoint Approximation

Approximate $\int_1^5 x\,dx$ using 4 rectangles of equal width, with heights determined by the right endpoint.

• $\Delta x = 1$

• Right endpoints: $x = 2, 3, 4, 5$

• Sum: $f(2) + f(3) + f(4) + f(5)$, each multiplied by $\Delta x$

Example 4: Application to Distance Traveled (Tabular Data)

Suppose Dr. Austin checks her velocity every hour while driving. The table below shows her velocity at several times:

Time (hours)	Velocity (mph)
0	0
1	58
2	62
3	63

To approximate the total distance traveled over 3 hours:

• Left endpoint method: Use the velocity at the start of each interval.

• Right endpoint method: Use the velocity at the end of each interval.

• Multiply each velocity by the time interval (1 hour) and sum the results.

Example Calculation:

• Left endpoints: $0 \times 1 + 58 \times 1 + 62 \times 1 = 120$ miles

• Right endpoints: $58 \times 1 + 62 \times 1 + 63 \times 1 = 183$ miles

Additional info: The actual distance is best approximated by increasing the number of intervals or using the midpoint method.