Question

Write down, but do not evaluate, the integral defined by the limit as \(\delta x \rightarrow 0\), or \(\delta y \rightarrow 0\), of the following sums: (a) \(\sum_{x=0}^{x=1} x^{3} \delta x\) (b) \(\sum_{x=0}^{x=4} 4 \pi x^{2} \delta x\) (c) \(\sum_{y=0}^{y=1} y^{3} \delta y\) (d) \(\sum_{x=0}^{x=1} 6 m x^{2} \delta x\)

Short answer

(a) \(\sum_{x=0}^{x=1} x^{3} \delta x\) (b) \(\sum_{x=0}^{x=4} 4 \pi x^{2} \delta x\) (c) \(\sum_{y=0}^{y=1} y^{3} \delta y\) (d) \(\sum_{x=0}^{x=1} 6 m x^{2} \delta x\) (a) The integral expression corresponding to the Riemann sum is \(\int_{0}^{1} x^{3} dx\). (b) The integral expression corresponding to the Riemann sum is \(\int_{0}^{4} 4 \pi x^{2} dx\). (c) The integral expression corresponding to the Riemann sum is \(\int_{0}^{1} y^{3} dy\). (d) The integral expression corresponding to the Riemann sum is \(\int_{0}^{1} 6mx^{2} dx\).

Step-by-step solution

Identify the Riemann sum

Here \(\sum_{x=0}^{x=1} x^{3} \delta x\) is a Riemann sum with function \(f(x) = x^{3}\).

Write down the integral form

To convert the Riemann sum into an integral, the sum becomes the integral and the \(\delta x\) turns into \(dx\). So we get: \(\int_{0}^{1} x^{3} dx\) (b)

Identify the Riemann sum

Here \(\sum_{x=0}^{x=4} 4 \pi x^{2} \delta x\) is a Riemann sum with function \(f(x) = 4 \pi x^{2}\).

Write down the integral form

To convert the Riemann sum into an integral, the sum becomes the integral and the \(\delta x\) turns into \(dx\). So we get: \(\int_{0}^{4} 4 \pi x^{2} dx\) (c)

Identify the Riemann sum

Here \(\sum_{y=0}^{y=1} y^{3} \delta y\) is a Riemann sum with function \(f(y) = y^{3}\).

Write down the integral form

To convert the Riemann sum into an integral, the sum becomes the integral and the \(\delta y\) turns into \(dy\). So we get: \(\int_{0}^{1} y^{3} dy\) (d)

Identify the Riemann sum

Here \(\sum_{x=0}^{x=1} 6 m x^{2} \delta x\) is a Riemann sum with function \(f(x) = 6mx^{2}\).

Write down the integral form

To convert the Riemann sum into an integral, the sum becomes the integral and the \(\delta x\) turns into \(dx\). So we get: \(\int_{0}^{1} 6mx^{2} dx\)

Key concepts

Definite Integral

The concept of a definite integral is central to understanding the relationship between sums and integrals in calculus. A definite integral represents the area under a curve on a graph within specified limits, typically expressed as \( \int_{a}^{b} f(x) dx \), where \( f(x) \) is a function and \( a \) and \( b \) are the lower and upper limits of the integral, respectively.

A key aspect of the definite integral is that it provides a way to calculate the accumulation of quantities, which can represent anything from areas to accumulated change. In the context of Riemann sums, as the interval widths \( \delta x \) or \( \delta y \) approach zero, the sum becomes more accurate, and in the limit, it defines the value of the definite integral.

Integral Calculus

Integral calculus is the branch of mathematics that deals with finding and analyzing integrals, both definite and indefinite. It is a fundamental part of calculus, along with differential calculus. Integral calculus focuses on accumulation functions, anti-differentiation, and the computation of volume and area using integrals.

Integral calculus studies ways to break down a complicated function into simpler parts to compute the area under a curve, the total distance traveled, or the total growth over time. Understanding the properties of Riemann sums is essential for mastering integral calculus because it sets the foundation for how we interpret the integral of functions over a given interval.

Mathematical Limits

Mathematical limits are a pivotal concept in calculus, involving the behavior of a function as its input approaches a certain value. Limits help in defining both derivatives and integrals. They provide the means to describe the value that a function approaches as the interval \( \delta x \) or \( \delta y \) becomes arbitrarily small.

For example, when we say \( \delta x \rightarrow 0 \) in a Riemann sum, we are using the concept of a limit to describe the process where the sum gets closer to the true value of the definite integral. This limit is what connects discrete sums of areas in Riemann sums to the continuous accumulation of area in definite integrals. An understanding of limits is therefore crucial to transition from sums to integrals.

Sums as Integrals

Considering sums as integrals involves the recognition that the summing of infinitely many infinitely small quantities can be represented by an integral. Riemann sums are a specific kind of sum used to approximate the area under a curve – the sum of products of function values at certain points and the width of intervals around those points.

As an example, in the textbook exercise, the Riemann sum \( \sum_{x=0}^{x=1} x^{3} \delta x \) can be seen as summing the areas of rectangles with height \( x^{3} \) and width \( \delta x \) between 0 and 1 on the x-axis. When \( \delta x \) becomes very small, this sum can be converted into the definite integral \( \int_{0}^{1} x^{3} dx \) by employing the concept of a limit. This interpretation of sums as integrals is fundamental in integral calculus, providing a bridge between discrete and continuous mathematics.