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Chapter 5: Problem 6

Evaluate the integrals $$ \int_{-2}^{3}\left(x^{3}-2 x+3\right) d x $$

Short Answer

Expert verified
26.25

Step by step solution

01

Break Down the Integral

The integral we need to evaluate is \( \int_{-2}^{3} (x^3 - 2x + 3) \, dx \). This integral can be split into three separate integrals: \( \int_{-2}^{3} x^3 \, dx \), \( \int_{-2}^{3} -2x \, dx \), and \( \int_{-2}^{3} 3 \, dx \). Solve each separately and then combine the results.
02

Integrate Each Term

First, integrate each term separately:- The integral of \( x^3 \) with respect to \( x \) is \( \frac{1}{4}x^4 + C \).- The integral of \( -2x \) is \( -x^2 + C \).- The integral of the constant \( 3 \) is \( 3x + C \).
03

Evaluate Definite Integrals

Plug the limits of integration into each indefinite integral and subtract:1. \( \int_{-2}^{3} x^3 \, dx = \left[\frac{1}{4}x^4\right]_{-2}^{3} = \frac{1}{4}(3^4) - \frac{1}{4}((-2)^4) \).2. \( \int_{-2}^{3} -2x \, dx = \left[-x^2\right]_{-2}^{3} = -(3^2) - (-(-2)^2) \).3. \( \int_{-2}^{3} 3 \, dx = \left[3x\right]_{-2}^{3} = 3(3) - 3(-2) \).
04

Calculate Each Value

Calculate the value of each definite integral:1. \( \int_{-2}^{3} x^3 \, dx = \frac{1}{4}(81) - \frac{1}{4}(16) = 20.25 - 4 = 16.25 \).2. \( \int_{-2}^{3} -2x \, dx = -9 - (-4) = -9 + 4 = -5 \).3. \( \int_{-2}^{3} 3 \, dx = 9 + 6 = 15 \).
05

Combine Results

Combine the results of the three integrals:- Sum the values: \( 16.25 + (-5) + 15 = 26.25 \).

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Key Concepts

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

Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a pivotal principle in mathematics that connects differentiation and integration. This theorem essentially states that if we have a continuous function defined on a closed interval, the process of differentiating an integral function returns the original function. When solving a definite integral, we typically follow these basic steps:
  • Find the antiderivative of the function, which is also known as the indefinite integral.
  • Evaluate this antiderivative at the boundaries of the interval.
  • Subtract to find the net "accumulation" within those boundaries.
In our given problem, we applied the fundamental theorem by first determining the indefinite integral of the polynomial function. Afterward, we used the upper and lower bounds of (-2 and 3) to find the total area represented by the definite integral.
Integration Techniques
Integration involves finding the antiderivative of a function, and there are various techniques to do this effectively. In this problem, the polynomial expression was broken down into simpler parts, making use of a basic integration method which is often sufficient for polynomials.For each term of the polynomial:
  • The power rule was used, which states that the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\).
  • Constants are integrated by multiplying them with \(x\) since the antiderivative of a constant \(C\) is \(Cx\).
  • The linearity of integration allows for breaking down complex expressions into simpler ones.
Here, these techniques were systematically applied by integrating each polynomial term separately before combining the results, ensuring clarity and accuracy in calculations.
Polynomial Integration
Polynomial integration is a straightforward process, but it's essential to carry it out with precision. This involves using simple algebraic rules, particularly the power rule for integration.For a polynomial function like \(x^3 - 2x + 3\):
  • Each term is integrated individually: \(x^3\), \(-2x\), and \(3\).
  • The antiderivative of \(x^3\) was calculated as \(\frac{1}{4}x^4\).
  • For \(-2x\), the antiderivative was \(-x^2\).
  • Finally, the constant \(3\) is integrated to \(3x\).
Every polynomial term contributes separately to the final integral solution, which is systematically summed up. This ensures that each component of the polynomial is accounted for, leading to an accurate calculation of the total definite integral within the specified limits.

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Most popular questions from this chapter

You will find the area between curves in the plane when you cannot find their points of intersection using simple algebra. Use a CAS to perform the following steps: a. Plot the curves together to see what they look like and how many points of intersection they have. b. Use the numerical equation solver in your CAS to find all the points of intersection. c. Integrate \(| f(x)-g(x)\) over consecutive pairs of intersection values. d. Sum together the integrals found in part (c). $$ f(x)=x+\sin (2 x), \quad g(x)=x^{3} $$

In Exercises \(95-98,\) use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100,200,\) and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=(\) average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$ f(x)=x \sin ^{2} \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right] $$

Use the Substitution Formula in Theorem 7 to evaluate the integrals. $$ \text { a. } \int_{-1}^{1} \frac{5 r}{\left(4+r^{2}\right)^{2}} d r \quad \text { b. } \int_{0}^{1} \frac{5 r}{\left(4+r^{2}\right)^{2}} d r $$

If you average 30 \(\mathrm{mi} / \mathrm{h}\) on a \(150-\mathrm{mi}\) trip and then return over the same 150 \(\mathrm{mi}\) at the rate of \(50 \mathrm{mi} / \mathrm{h},\) what is your average speed for the trip? Give reasons for your answer.

Evaluate the integrals in Exercises \(17-50\) $$ \int \frac{1}{x^{3}} \sqrt{\frac{x^{2}-1}{x^{2}}} d x $$
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