Finite approximations and indefinite integral

Mathematical Thinking I

Finite approximations and indefinite integral

Elaboración: Dra. Diana Denys Jiménez Suro Dr. Antonio Jesús Sánchez Hernández Campus Estado de México Campus Santa Fe

The indefinite integral

The integral is one of the key tools in calculus, and its use applies to the calculation of important quantities in mathematics and science, such as areas, volumes, and lengths of curved paths, to mention just a few. The idea behind the integral is that it is possible to calculate the mentioned quantities by breaking them down into infinitesimal parts and summing the contributions of each one.

Area and its estimation using finite sums

Suppose we need to determine the area of the shaded region 𝑅 that is above the x-axis, below the graph of 𝑦=1−𝑥2, and between the vertical lines 𝑥=0 and 𝑥=1.

Unfortunately, there isn't a simple geometric formula for calculating the areas of general shapes with curved boundaries, such as the region 𝑅. So, how can we determine the area of 𝑅? Although we don't have a method to find the exact area of R, it is possible to approximate it in a straightforward way.

Overestimation of the area of 𝑅 using two and four rectangles that contain 𝑅

Overestimation using 2 rectangles

In the first figure, two rectangles are shown that, together, contain the region 𝑅. Each rectangle has a width of 1/2, and they have heights (from left to right) of 1 and 3/4. The height of each rectangle is the maximum value of the function 𝑓, obtained by evaluating 𝑓 at the left endpoint of the subinterval in [0, 1] that forms the base of the rectangle. The first estimate of the area of the region 𝑅 using these two rectangles is:

Overestimation using 4 rectangles

The previous estimate is greater than the exact area of 𝐴 because the two rectangles contain 𝑅. We say that 0.875 is an upper sum, as it is obtained by taking the height of each rectangle as the maximum (larger) value of 𝑓(𝑥) for a point 𝑥 in the interval that forms the base of the rectangle. In the second figure, we improve the estimate using four narrower rectangles, each with a width of 1/4, which together contain the region 𝑅. The four rectangles provide the approximation:

Upper sum of the area of 𝑅 using 16 rectangles that contain 𝑅

Now, suppose that to estimate the area of 𝑅, we use rectangles contained within the region. By summing these rectangles with a height equal to the minimum value of 𝑓(𝑥) for a point 𝑥 in each subinterval forming the base, we obtain a lower sum that approximates the area.

Lower sum of the area of 𝑅 using 16 rectangles contained within 𝑅

Finite approximations

It's clear that the actual value of the area 𝐴 lies between the lower sum and the upper sum. For different partitions of rectangles, we have:

Desmos

Antiderivative.

If we have a function 𝑓(𝑥), we say that 𝐹(𝑥) is an antiderivative or primitive of 𝑓(𝑥) if it holds for every 𝑥 in an interval 𝐼.

Indefinite integral

When we have a function 𝑓(𝑥) with an antiderivative 𝐹(𝑥), we can observe that 𝐹(𝑥)+𝐶 is also an antiderivative of 𝑓(𝑥). Based on this fact, we refer to the indefinite integral of the function 𝑓(𝑥) as

Basic indefinite integrals

Properties of the integral.

If we have two functions 𝑓(𝑥) and 𝑔(𝑥), and a constant 𝑐, the following properties hold: One must be cautious because, unlike derivatives, there are no formulas for the product or quotient of functions.

Exercises

References

Thomas

Calculus Early Transcendentals13a. Edición, PEARSON

2015

Thomas

Cálculus Multivariable13a. Edición, PEARSON

2015

Stewart

Calculus Early Transcendentals8a. Edición, CENGAGE

2018

Teachers

Antonio Jesús Sánchez Hernández Campus Santa Fe ajsanchez@tec.mx

Diana Denys Jiménez Suro Campus Estado de México ddjimenez@tec.mx