/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Evaluate the integrals. $$ \... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 9

Evaluate the integrals. $$ \int_{0}^{1}\left(2.1 x-4.3 x^{1.2}\right) d x $$

Short Answer

Expert verified
The integral \( \int_{0}^{1}\left(2.1 x-4.3 x^{1.2}\right) d x \) can be found by first calculating the antiderivatives using the power rule, then combining the antiderivatives, applying the Fundamental Theorem of Calculus, and simplifying. The final result is \(-0.9045\).

Step by step solution

01

Find the antiderivatives of each term

First, we need to find the antiderivative of each term in the integrand. The antiderivative of a term with respect to x can be found using the power rule: \[ \int x^n dx = \frac{x^{n+1}}{n+1} \] Using the power rule for each term, we have: \[ \int 2.1x dx = 2.1 \int x^{1} dx = \frac{2.1x^{1+1}}{1+1} \] and \[ \int -4.3x^{1.2} dx = -4.3 \int x^{1.2} dx = \frac{-4.3x^{1.2+1}}{1.2+1} \]
02

Combine the antiderivatives

Now we need to combine the antiderivatives we found above: \[ \int_{0}^{1}\left(2.1 x-4.3 x^{1.2}\right) d x = \frac{2.1x^{2}}{2} - \frac{4.3x^{2.2}}{2.2} \]
03

Apply the Fundamental Theorem of Calculus

Now we have to apply the interval that we are finding the integral over: [0, 1]. By using the Fundamental Theorem of Calculus, we will evaluate the antiderivative at the upper limit (1) and subtract the value of the antiderivative evaluated at the lower limit (0): \[ \left[\frac{2.1x^{2}}{2} - \frac{4.3x^{2.2}}{2.2}\right]_{0}^{1} = \left(\frac{2.1(1)^{2}}{2} - \frac{4.3(1)^{2.2}}{2.2}\right) - \left(\frac{2.1(0)^{2}}{2} - \frac{4.3(0)^{2.2}}{2.2}\right) \]
04

Simplify and find the final answer

We just need to simplify the expression to find the result: \[ \left(\frac{2.1}{2} - \frac{4.3}{2.2}\right) - \left(\frac{0}{2} - \frac{0}{2.2}\right) = 1.05 - 1.9545 = -0.9045 \] So the integral is equal to -0.9045.

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Key Concepts

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

Antiderivative
Finding an antiderivative means determining the original function whose derivative would result in the integrand we have been given. In essence, if you have a function and you take its derivative, you'll get the derivative function. When you perform the opposite operation, you find the antiderivative.
  • Given a function like \( f(x) = 2.1x - 4.3x^{1.2} \), finding its antiderivative with respect to \( x \) involves reversing the derivative process.
  • For polynomials and terms like \( x^n \), we can use the power rule of integration, which is a direct tool for finding antiderivatives efficiently.
The antiderivative is a key first step in solving definite integrals, as it helps set the stage for further evaluation through definite limits.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus serves as the bridge between differentiation and integration. This principle states two remarkable things: the relationship between the derivative and the integral of a function, and how to evaluate a definite integral using antiderivatives.
  • Part 1 connects the concept of an antiderivative with that of a definite integral, showing that the process of integrating a function is essentially the reverse of differentiating it.
  • Part 2 affirms that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), the integral of \( f \) over \([a, b]\) is given by \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \].
This theorem provides a systematic approach to evaluating definite integrals using boundary values.
Power Rule
The power rule for integration is a reliable technique to find the antiderivative of terms in the form \( x^n \). When the function is expressed as a polynomial, applying the power rule becomes straightforward and invaluable.
  • The rule states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
  • When using this rule during definite integration, constants of integration cancel out since they do not affect the final result, only finite changes (from \( a \) to \( b \)) are considered.
Mastering the power rule simplifies many integration problems especially when dealing with polynomials or power functions.
Integral Evaluation
Evaluating a definite integral means computing the antiderivative's values at specified bounds and subtracting the result of the lower bound from the upper bound. This task often utilizes the Fundamental Theorem of Calculus.
  • E.g., for \( \int_{0}^{1} \, (2.1x - 4.3x^{1.2}) \, dx \), the antiderivatives are first summed up: \( \frac{2.1x^2}{2} - \frac{4.3x^{2.2}}{2.2} \), then evaluated at 1 and 0.
  • Finally, calculate \( F(1) - F(0) \) to get the net area under the curve over the interval \([0,1]\).
Simplification and careful numerical computation gives the final answer. In this case, the result was \(-0.9045\), indicating the total area with respect to the x-axis encompassed within those limits.

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