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Chapter 1: Problem 135

Suppose that for each \(i\) such that \(1 \leq i \leq N\) one has \(\quad \int_{i-1}^{i} f(t) d t=i^{2} . \quad\) Show \(\quad\) that \(\int_{0}^{N} f(t) d t=\frac{N(N+1)(2 N+1)}{6}\)

Short Answer

Expert verified
The integral \(\int_{0}^{N} f(t) dt\) equals \(\frac{N(N+1)(2N+1)}{6}\) by summing squares.

Step by step solution

01

Understand the Problem

We are given that, for each segment \([i-1, i]\), the integral of \(f(t)\) is equal to \(i^2\). Our goal is to find the integral from 0 to \(N\) and show that it equals \(\frac{N(N+1)(2N+1)}{6}\).
02

Break Down the Integral

Consider the integral \(\int_{0}^{N} f(t) dt\). This integral can be broken down into a sum of individual integrals over each integer interval: \(\int_{0}^{1} f(t) dt + \int_{1}^{2} f(t) dt + \cdots + \int_{N-1}^{N} f(t) dt\), which corresponds to the sum \(1^2 + 2^2 + \cdots + N^2\).
03

Recognize the Sum of Squares Formula

The sum of squares formula is given by \(1^2 + 2^2 + \cdots + N^2 = \frac{N(N+1)(2N+1)}{6}\). This is a well-known formula for the sum of squares of the first \(N\) natural numbers.
04

Apply the Sum of Squares Formula

Since each integral \(\int_{i-1}^{i} f(t) dt = i^2\), it follows that \(\int_{0}^{N} f(t) dt = 1^2 + 2^2 + \cdots + N^2\). Applying the sum of squares formula gives \(\int_{0}^{N} f(t) dt = \frac{N(N+1)(2N+1)}{6}\).
05

Conclusion

Having applied the sum of squares formula to the sum of integrals over each sub-interval, we conclude that \(\int_{0}^{N} f(t) dt = \frac{N(N+1)(2N+1)}{6}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus
Integral calculus is a fundamental branch of mathematics concerned with finding the accumulation of quantities. At its core, it focuses on calculating integrals, which can be understood as the area under a curve within a specified interval. In the context of this problem, we have the integral of a function, denoted as \(f(t)\), over various intervals from \(i-1\) to \(i\). The cumulative result from \(0\) to \(N\) gives us insight into the behavior of the function over this entire range.
  • It involves calculating areas which help in predicting total accumulated values over time or space.
  • The practice forms the foundation for understanding further core calculus concepts like derivatives and the Fundamental Theorem of Calculus.
  • Integral calculus helps in various applied fields such as physics, engineering, and economics.
Sum of Squares
The sum of squares is a straightforward yet crucial arithmetic formula used across many mathematical disciplines. It sums the squares of consecutive integers, a technique often utilized in sequences and series. For this problem, the sum \(1^2 + 2^2 + \, \ldots \, + N^2\) can be calculated using the formula \(\frac{N(N+1)(2N+1)}{6}\). This formula simplifies computations and solves the integral challenge by providing a neat expression for the sum:
  • Each \(i^2\) represents the value of the integral for each segment.
  • Summing these values gives the total integral over the entire interval from 0 to \(N\).
  • This formula is a foundational element in algebra, frequently used in statistics and data analysis to measure variability.
Definite Integrals
In calculus, definite integrals are a way to find the area under a curve between two specified points. They provide a numerical value representing this accumulated area. Unlike indefinite integrals, which are more about finding a family of functions, definite integrals result in a specific number. Here's how they play into our problem:
  • Each interval \([i-1, i]\) yields a definite integral represented as \(i^2\).
  • The role of definite integrals here is to express the integral \(\int_0^N f(t) dt\) as a summation of these individual segments.
  • Definite integrals help in understanding total quantities, like distance, mass, or probability, depending on the context.
Mathematical Proof
A mathematical proof is a logical argument verifying the truth of a statement. In this exercise, the proof involves logical reasoning to show that the integral from 0 to \(N\) results in a sum of squares expressed by the sum \(\frac{N(N+1)(2N+1)}{6}\). To establish this proof, we need:
  • Understanding of the problem setup, where \(\int_{i-1}^i f(t) dt = i^2\).
  • Breaking the overall integral into a sum of known segments \(1^2 + 2^2 + ... + N^2\).
  • Application of the sum of squares formula to verify and prove the given integral equation.
The logical structure must ensure each step follows from previous ones, maintaining accuracy and clarity. This type of rigorous demonstration is crucial in mathematics.

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Most popular questions from this chapter

In the following exercises, find the antiderivative using the indicated substitution. $$ \int(3 x-2)^{-11} d x ; u=3 x-2 $$

In the following exercises, integrate using the indicated substitution. $$ \int \frac{\sin x+\cos x}{\sin x-\cos x} d x ; u=\sin x-\cos x $$

In the following exercises, integrate using the indicated substitution. $$ \int \frac{x}{x-100} d x ; u=x-100 $$

In the following exercises, integrate using the indicated substitution. $$ \int \frac{1-x^{2}}{3 x-x^{3}} d x ; u=3 x-x^{3} $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t \sin \left(t^{2}\right) \cos \left(t^{2}\right) d t $$
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