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Chapter 5: Problem 6

$$\text { Evaluate } \int_{0}^{2} 3 x^{2} d x \text { and } \int_{-2}^{2} 3 x^{2} d x$$

Short Answer

Expert verified
Answer: The values of the definite integrals are 8 and 16, respectively.

Step by step solution

01

Find the antiderivative of 3x^2

Start by finding the antiderivative of 3x^2. When finding the antiderivative, we need to apply the power rule, which states that $$\int x^n dx = \frac{1}{n+1}x^{n+1}$$ So, applying the power rule to 3x^2, we get: $$\int 3x^2 dx = 3\int x^2 dx = 3\frac{1}{3}x^{3} = x^3$$
02

Evaluate the first integral, $$\int_{0}^{2} 3x^2 dx$$

Now that we have found the antiderivative, we can find the definite integral by evaluating the antiderivative at the upper and lower limits and subtracting the two results: $$\int_{0}^{2} 3x^2 dx = [x^3]_{0}^{2} = (2^3 - 0^3) = 8$$
03

Evaluate the second integral, $$\int_{-2}^{2} 3x^2 dx$$

Similar to the previous step, evaluate the antiderivative at the upper and lower limits and subtract the two results: $$\int_{-2}^{2} 3x^2 dx = [x^3]_{-2}^{2} = (2^3 - (-2)^3) = 8 - (-8) = 16$$ So, the values for the two definite integrals are: 1. $$\int_{0}^{2} 3x^2 dx = 8$$ 2. $$\int_{-2}^{2} 3x^2 dx = 16$$

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Key Concepts

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

Antiderivative
The concept of the antiderivative is fundamental in calculus. When we talk about finding an antiderivative, we essentially mean finding a function whose derivative gives us the original function.
In the exercise, we are given the function \(3x^2\). To find its antiderivative, we have to determine a function that, when differentiated, returns \(3x^2\).
The antiderivative of \(3x^2\) is derived by applying the power rule, resulting in \(x^3\).
  • Antiderivatives are also known as indefinite integrals.
  • The process of finding an antiderivative involves reversing differentiation.
  • In this case, we check our work by taking the derivative of \(x^3\) and verifying it equals \(3x^2\).
Understanding how to perform the reverse operation of differentiation is crucial for solving definite integrals and finding areas under curves.
Power Rule
The power rule is a basic tool used in calculus to find the derivatives and antiderivatives of power functions. When dealing with integrals like \( \int x^n dx \), the power rule tells us that the antiderivative is \( \frac{1}{n+1}x^{n+1} + C \), where \(C\) is a constant.
The rule makes integration of polynomials intuitive and straightforward. For our specific exercise, we have \(3x^2\), and using the power rule for integration:
  • Identify the power \(n=2\) for \(x^2\).
  • Apply the formula: \(\frac{1}{2+1}x^{2+1} = \frac{1}{3}x^3\).
  • Multiply by the constant 3 to get \(x^3\).
Understanding the power rule simplifies many integration problems and is essential for evaluating integrals efficiently.
Symmetry in Integration
Integration can often reveal important symmetries in the functions we deal with. In this exercise, we looked at two definite integrals: \(\int_{0}^{2} 3x^2 dx \) and \(\int_{-2}^{2} 3x^2 dx\).
Symmetrical properties simplify calculations or provide insights into the behavior of a function over an entire interval.
  • The first integral from \(0\) to \(2\) uses the antiderivative evaluated at its limits, leading to a straightforward calculation.
  • Interestingly, the symmetry of \(3x^2\) about the y-axis means the integral from \(-2\) to \(2\) is just double that of from \(0\) to \(2\).
  • This occurs because \(x^2\) is an even function, and integrating an even function symmetrically around zero yields twice the result of integrating from zero to the endpoint.
Grasping these symmetrical properties allows for deeper insights and better problem-solving strategies in calculus.

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Most popular questions from this chapter

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $$\int_{1}^{e^{2}} \frac{\ln p}{p} d p$$

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. $$\int\left(x^{3 / 2}+8\right)^{5} \sqrt{x} d x$$

Variations on the substitution method Evaluate the following integrals. $$\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$$

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $$\int_{0}^{\pi / 4} \frac{\sin x}{\cos ^{2} x} d x$$

Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator. The midpoint Riemann sum for \(f(x)=x^{3}\) on [3,11] with \(n=32\)
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