Integrals

Section 4.1 Integrals

Standards

Demonstrate understanding of parts of definition of integral using an application

One of the motivations for the concept of integration is calculating areas. We know how to calculate areas of discrete shapes like rectangles, parallelograms, triangles, and from these we can figure out others. A goal is to calculate areas of continuously changing shapes like circles. If we are willing to accept approximations we can use discrete shapes to approximate the continuously changing shapes. Limits then provide a means to transition from appoximations to exact calculations.

🔗

Subsection 4.1.1 How Square Pegs Fill Round(ed) Holes

🔗

Activity 4.1.1. Comparing Approximations.

🔗

The goal of this activity is to compare the values of area approximations to actual areas and to compare approximations to each other.

🔗

The function is

f (x) = 2 x - x^{2} .

🔗

(a)

🔗

For each image in Figure 4.1.1, calculate the total area of the gray rectangles.

🔗

(b)

🔗

For all of the images in Figure 4.1.1, compare the area between the parabola and the

x

-axis and the total area of the gray rectangles in that image.

🔗

(c)

🔗

For each image in Figure 4.1.2, calculate the total area of the gray rectangles.

🔗

(d)

🔗

For each image in Figure 4.1.2, compare the area between the parabola and the

x

-axis and the total area of the gray rectangles in that image.

🔗

(e)

🔗

As the number of rectangles increased in Figure 4.1.1 how did the approximation change? You can use Activity 4.1.1 to experiment. There are two major parts to this answer.

🔗

(f)

🔗

As the number of rectangles increased in Figure 4.1.2 how did the approximation change? You can use Activity 4.1.1 to experiment. There are two major parts to this answer.

🔗