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Chapter 4: Problem 71

Proof Prove that \(\int_{a}^{b} x d x=\frac{b^{2}-a^{2}}{2}\).

Short Answer

Expert verified
The proof shows that \(\int_{a}^{b} x d x=\frac{b^{2}-a^{2}}{2}\), based on the Fundamental Theorem of Calculus.

Step by step solution

01

Statement of Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus connects differentiation and integration. It states: \(\int_{a}^{b} f'(x) d x = f(b) - f(a)\), where \(f'(x)\) is the derivative of \(f(x)\), and \(a\) and \(b\) are the limits of integration.
02

Apply Fundamental Theorem on the given Integral.

An antiderivative of the function \(x\) can be written as \(f(x) = \frac{x^2}{2}\). Now applying the Fundamental theorem of calculus to \(\int_{a}^{b} x d x\), so we have \(f(b) - f(a) = \frac{b^2}{2} - \frac{a^2}{2}\).
03

Simplify Expression

Simplify \(f(b) - f(a) = \frac{b^2}{2} - \frac{a^2}{2}\) to obtain \(f(b) - f(a) = \frac{b^2 - a^2}{2}\). So it is proved that \(\int_{a}^{b} x d x=\frac{b^{2}-a^{2}}{2}\).

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Key Concepts

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

Definite Integrals
Definite integrals calculate the accumulation of quantities, bounded by two points on the x-axis. They essentially measure the net area between the curve of a function and the x-axis, from one point to another. For example, in the exercise, the integral \( \int_{a}^{b} x \, dx \) computes the area beneath the curve of the function \( x \) between the limits \( a \) and \( b \). The result of this operation provides a single, specific value.

Understanding definite integrals is crucial because:
  • They summarize the total accumulated change.
  • Offer a clear geometric interpretation as areas under curves.
  • Allow calculation of exact measures given specific boundaries.
Antiderivatives
The antiderivative, also known as the indefinite integral, is a function whose derivative gives the original function. To solve a definite integral, one must first determine the antiderivative of the integrand, the function being integrated.

In this exercise, the integrand is \(x\). Its antiderivative can be represented as \( f(x) = \frac{x^2}{2} \). The role of antiderivatives is significant because they:
  • Help reverse the process of differentiation.
  • Offer insights into accumulating quantities continuously.
  • Provide the foundation needed to evaluate definite integrals.
By determining the antiderivative, you essentially find the function that describes the area accumulation, which is vital for computing definite integrals later on.
Integration Techniques
Integration techniques are strategies used to find antiderivatives and evaluate integrals more efficiently. For simple functions like \(x\), basic integration rules can be applied directly. However, more complex functions might require substitution, integration by parts, or partial fractions.

Key integration techniques include:
  • Basic Power Rule of Integration: Used for integrating polynomials. For example, the integral of \(x^n\) becomes \(\frac{x^{n+1}}{n+1}\) for \(n eq -1\).
  • Substitution: Simplifies the integral by substituting variables, making integration straightforward.
  • Integration by Parts: Based on the product rule, useful for integrating products of functions.
These techniques allow mathematicians and students to tackle a wide variety of functions, transforming a challenging integration task into a manageable problem.

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Most popular questions from this chapter

Depreciation The rate of depreciation \(d V / d t\) of a machine is inversely proportional to the square of \((t+1),\) where \(V\) is the value of the machine \(t\) years after it was purchased. The initial value of the machine was \(\$ 500,000,\) and its value decreased \(\$ 100,000\) in the first year. Estimate its value after 4 years.

Finding Values Find the constants \(a\) and \(b\) that maximize the value of $$ \int_{a}^{b}\left(1-x^{2}\right) d x $$ Explain your reasoning.

Upper and Lower Sums In your own words and using appropriate figures, describe the methods of upper sums and lower sums in approximating the area of a region.

True or False? In Exercises \(63-68\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The value of \(\int_{2}^{2} \sin \left(x^{2}\right) d x\) is 0

Approximating Area with the Midpoint Rule In Exercises \(61-64,\) use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of the function and the \(x\) -axis over the given interval. $$ f(x)=x^{2}+3, \quad[0,2] $$
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