Applications of Integration: Area and the Definite Integral

Ch. 6: Overview of Applications of Integration

A definite integral represents the total amount of a quantity, obtained by adding infinitely many small amounts of that quantity over an interval. In calculus, definite integrals are used to compute several different quantities:

• Area between curves (6.1)

• Volumes of solids of revolution (6.2 & 6.3)

• Arc length (6.4)

• Area of a surface of revolution (Surface Area, 6.5)

• Work (6.6)

• Hydrostatic force (6.7)

• Moments & centroids (center of mass, 6.8)

6.1 Area Between Curves

Estimating Area Using Rectangles

To estimate the area under a curve $y = f(x)$ on the interval $[a, b]$, we can approximate the region by dividing it into $n$ rectangles and summing their areas. The more rectangles used, the better the approximation.

• Left endpoint approximation: Use the left side of each subinterval to determine the rectangle's height.

• Right endpoint approximation: Use the right side of each subinterval for the height.

• Midpoint approximation: Use the midpoint of each subinterval for the height.

The total area is approximated by:

$A \approx A_1 + A_2 + \cdots + A_n$

Riemann Sums

The sum of the areas of the rectangles is called a Riemann sum:

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

where $x_i^*$ is a sample point in the $i$th subinterval and $\Delta x = \frac{b-a}{n}$.

Definite Integral as a Limit

The exact area under the curve is found by taking the limit as $n \to \infty$:

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

This limit is called the definite integral of $f$ from $a$ to $b$:

$\int_a^b f(x)\,dx$

The integral symbol $\int$ is an elongated S, representing "sum." The process of finding the area under a curve using limits of Riemann sums is called integration.

6.2 Definite Integrals

Definition of the Definite Integral

The definite integral of $f$ from $a$ to $b$ is defined as:

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

where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is any sample point in the $i$th subinterval $[x_{i-1}, x_i]$.

• The definite integral gives a numerical value (like $5$or$-3$).

• An indefinite integral gives an antiderivative (a function).

Geometric Interpretation

The definite integral $\int_a^b f(x)\,dx$ represents the net area between the graph of $f(x)$ and the $x$-axis on $[a, b]$.

6.3 The Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculus connects differentiation and integration:

If $F$ is any antiderivative of $f$ on $[a, b]$, then:

$\int_a^b f(x)\,dx = F(b) - F(a)$

• To evaluate a definite integral, find an antiderivative $F$ of $f$, then compute $F(b) - F(a)$.

6.4 Area Between Two Curves

General Formula

The area between two curves $y = f(x)$ (upper) and $y = g(x)$ (lower) on $[a, b]$ is:

$A = \int_a^b [f(x) - g(x)]\,dx$

This can be interpreted as summing the areas of infinitely many thin rectangles of height $f(x) - g(x)$ and width $dx$.

Area Between Curves (in terms of $y$)

If the equations are solved for $x$ in terms of $y$, the area between $x = f(y)$ (right) and $x = g(y)$ (left) from $y = c$ to $y = d$ is:

$A = \int_c^d [f(y) - g(y)]\,dy$

Steps for Finding Area Between Curves

1. Find the intersection points of the curves.

2. Graph the region.

3. Set up the integral(s) and evaluate.

Note: If the upper and lower functions switch, split the integral at the intersection point.

Examples

• Find the area of the region bounded by $y = x^2$, $y = x + 2$, and $x = 2$.

• Find the area between $y = \frac{3}{x}$ and $x = y^2 + 8$.

• Find the area between $y = 5\sin x$ and $y = 1 - 10\cos x$, with $0 \leq x \leq \frac{\pi}{2}$.

Additional info: These examples require setting up the appropriate definite integrals, possibly solving for intersection points, and integrating the difference of the functions over the specified interval.