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Chapter 4: Problem 58

Show that \(\int_{a}^{b} x^{2} d x=\frac{1}{3}\left(b^{3}-a^{3}\right)\).

Short Answer

Expert verified
Applying the power rule for integration, we find the antiderivative of \(x^2\) to be \(\frac{x^3}{3} + C\). Using the fundamental theorem of calculus and evaluating the limits of integration b and a, we obtain the final solution: \(\int_{a}^{b} x^2 dx = \frac{1}{3}\left(b^3 - a^3\right)\).

Step by step solution

01

Finding the indefinite integral (antiderivative) of the function

To find the indefinite integral of the function \(x^2\), we will use the power rule formula for integration: \[\int x^n dx = \frac{x^{n+1}}{n+1} + C\] where n is the power of x and C is the constant of integration. In our case, n = 2.
02

Applying the power rule to find the antiderivative of \(x^2\)

Applying the power rule, we have: \[\int x^2 dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C\]
03

Applying the fundamental theorem of calculus

Now that we have the antiderivative of the function \(x^2\), we can apply the fundamental theorem of calculus: \[\int_{a}^{b} x^2 dx = \left[\frac{x^3}{3} + C\right]_a^b = \left(\frac{b^3}{3} + C\right) - \left(\frac{a^3}{3} + C\right)\]
04

Finalizing the solution

Simplifying the expression we have: \[\int_{a}^{b} x^2 dx = \frac{b^3}{3} + C - \frac{a^3}{3} - C = \frac{1}{3}\left(b^3 - a^3\right)\] Hence, we have shown that \(\int_{a}^{b} x^2 dx = \frac{1}{3}\left(b^3 - a^3\right)\).

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Key Concepts

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

Antiderivative
An antiderivative, often referred to as an indefinite integral, is a function whose derivative results in the given original function. Finding an antiderivative is the reverse process of differentiation.
For instance, if we take the function \(x^2\), its antiderivative is the function that, when differentiated, gives back \(x^2\).
The notation for an indefinite integral is \(\int x^n \, dx\), and the result includes a constant of integration \(C\), because antiderivatives are not unique. This is due to the fact that the derivative of a constant is zero. In the context of our exercise, the antiderivative of \(x^2\) is \(\frac{x^3}{3} + C\).
  • Finding an antiderivative is akin to "going backwards" from derivatives.
  • An indefinite integral is a family of functions, all differing by a constant.
  • The technique used in our problem is the power rule for antiderivatives, resulting in \(\frac{x^3}{3} + C\).
Power Rule
The Power Rule is a fundamental technique used in both differentiation and integration.
For differentiation, it states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). When used for integration, it helps us find antiderivatives.
To integrate using the Power Rule, the formula is \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n\) cannot be \(-1\).
In our specific example of integrating \(x^2\), we apply the Power Rule to increase the exponent by one, making it \(x^3\), and then divide by this new exponent.
  • The resulting antiderivative for \(x^2\) becomes \(\frac{x^3}{3} + C\).
  • This rule simplifies the process of integration considerably for polynomials.
Remember that while the Power Rule is very useful, it applies straightforwardly only when dealing with polynomial functions.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a critical link between differentiation and integration.
It consists of two main parts, both essential in solving definite integrals:
  • The first part states that if \(F\) is an antiderivative of \(f\) on an interval \([a, b]\), then \(\int_a^b f(x) \, dx = F(b) - F(a)\).
  • The second part assures that every continuous function has an antiderivative, implying that integration is feasible.
In our problem, this theorem allows us to evaluate the definite integral \(\int_{a}^{b} x^2 \, dx\) by using the antiderivative \(F(x) = \frac{x^3}{3}\) obtained earlier. We evaluate it from \(a\) to \(b\), leading us to the expression \(\frac{1}{3}(b^3 - a^3)\).
The subtracting of values \(F(b) - F(a)\) showcases how the Fundamental Theorem provides a way to calculate the precise accumulation of change across intervals.

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