5.1 Overview & Applications of Integration

Introduction to Definite Integrals

A definite integral represents the total amount of a quantity, obtained by adding infinitely many small amounts of that quantity over an interval. This concept is foundational in calculus and is used to compute areas, volumes, and other quantities that accumulate continuously.

• Area between curves (5.1, 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 & center of mass/centroid (6.8)

5.1 Area as a Limit; Riemann Sums

Estimating Area Under a Curve

To estimate the area under the graph of y = f(x) on the interval [a, b], we can approximate the region using rectangles. The sum of the areas of these rectangles gives an approximation to the total area.

• Divide the interval [a, b] into n subintervals of equal width Δx.

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

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

• As n increases (rectangles become thinner), the approximation improves.

• The exact area is obtained in the limit as n approaches infinity.

5.2 Definite Integrals

Definition and Notation

The definite integral of a function f(x) from a to b is defined as the limit of Riemann sums:

• Δx is the width of each subinterval:

• x_i^* is a sample point in the ith subinterval.

• The integral symbol is an elongated S, representing summation.

The definite integral gives a numerical value (a number), representing the net area under the curve between a and b.

Properties of Definite Integrals

• The definite integral is independent of the choice of sample points as n approaches infinity.

• If f(x) is continuous on [a, b], the definite integral exists.

5.3 The Fundamental Theorem of Calculus (FTC)

Connecting Differentiation and Integration

The Fundamental Theorem of Calculus links the process of integration with differentiation. It states:

If is an antiderivative of on [a, b], then:

• To evaluate a definite integral, find an antiderivative of , then compute .

6.1 Area Between Two Curves

Finding the Area Between Curves

The area between two curves (upper) and (lower) on the interval [a, b] is given by:

• If both equations are solved for x in terms of y, the area is:

Steps to Find Area Between Curves

1. Find the intersection points of the curves.

2. Graph the functions to determine which is on top (or right, if integrating with respect to y).

3. Set up the definite integral and evaluate.

Examples

• Example 1: Find the area of the region bounded by , , and .

• Example 2: Find the area between and , .

• Example 3: Find the area between and , with .

Additional info: The notes also include graphical illustrations of Riemann sums, the process of taking limits, and the geometric interpretation of definite integrals as areas under or between curves.