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Chapter 5: Problem 101

In the following exercises, given that \(\int_{0}^{1} x d x=\frac{1}{2}, \int_{0}^{1} x^{2} d x=\frac{1}{3}, \quad\) and \(\quad \int_{0}^{1} x^{3} d x=\frac{1}{4}\) compute the integrals. \(\int_{0}^{1}(1-2 x)^{3} d x\)

Short Answer

Expert verified
The integral is 0.

Step by step solution

01

Expand the Expression

First, expand \((1 - 2x)^3\) using the binomial theorem. The binomial theorem states \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = 1\), \(b = -2x\), and \(n = 3\).\[(1 - 2x)^3 = \binom{3}{0}(1)^3(2x)^0 - \binom{3}{1}(1)^2(2x)^1 + \binom{3}{2}(1)^1(2x)^2 - \binom{3}{3}(1)^0(2x)^3\]Simplifying gives:\[(1 - 2x)^3 = 1 - 6x + 12x^2 - 8x^3\]
02

Set Up Integrals

Now, find the integral of each term from the expansion over the interval [0, 1]:\[\int_{0}^{1} (1 - 6x + 12x^2 - 8x^3) \, dx = \int_{0}^{1} 1 \, dx - 6 \int_{0}^{1} x \, dx + 12 \int_{0}^{1} x^2 \, dx - 8 \int_{0}^{1} x^3 \, dx\]
03

Evaluate the Integrals

Evaluate each integral separately:1. \( \int_{0}^{1} 1 \, dx = 1\) because the integral of a constant over \([0, 1]\) is the constant times the length of the interval.2. \(- 6 \int_{0}^{1} x \, dx = -6 \times \frac{1}{2} = -3\)3. \(12 \int_{0}^{1} x^2 \, dx = 12 \times \frac{1}{3} = 4\)4. \(- 8 \int_{0}^{1} x^3 \, dx = -8 \times \frac{1}{4} = -2\)
04

Combine the Results

Add the results from Step 3 together:\[1 - 3 + 4 - 2\]Evaluate the expression:\[1 - 3 + 4 - 2 = 0\]
05

Conclusion

The value of the integral \(\int_{0}^{1}(1-2x)^{3} \, dx\) is \(0\).

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Key Concepts

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

Binomial Theorem
The binomial theorem is a vital tool in algebra that helps expand expressions of the form \((a+b)^n\). This theorem allows us to represent the expansion as a sum of terms formed with coefficients called binomial coefficients. To use the binomial theorem:
  • Identify your values of \(a\), \(b\), and \(n\) in the expression \((a+b)^n\).
  • Use the formula: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
  • The binomial coefficient \(\binom{n}{k}\) is calculated using combinations, which represent the different ways to choose \(k\) items out of \(n\), and is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
With these steps, you can systematically expand any binomial raised to a positive integer power. It's especially useful when dealing with integral calculus, where expanded forms simplify the integration process.
Polynomial Expansion
When you expand a binomial expression, you transform it into a polynomial, which is a sum of terms with different powers of a variable. Take for example, \((1 - 2x)^3\):
  • Use the binomial theorem to express it as: \[(1 - 2x)^3 = \binom{3}{0}(1)^3(-2x)^0 - \binom{3}{1}(1)^2(-2x)^1 + \binom{3}{2}(1)^1(-2x)^2 - \binom{3}{3}(1)^0(-2x)^3\]
  • Simplify to get:\[1 - 6x + 12x^2 - 8x^3\]
Each step involves identifying coefficients and restructuring terms to get a polynomial form. This expression is now ready for operations like differentiation or integration, making these processes much more straightforward.
Integral Calculation
The process of integrating a function involves finding the area under a curve represented by that function. In our example, the interval is [0, 1]. We have four separate integrals to evaluate:
  • The constant term: \[\int_{0}^{1} 1 \, dx = 1\]
  • The linear term: \[- 6 \int_{0}^{1} x \, dx = -6 \times \frac{1}{2} = -3\]
  • The quadratic term:\[12 \int_{0}^{1} x^2 \, dx = 12 \times \frac{1}{3} = 4\]
  • The cubic term:\[- 8 \int_{0}^{1} x^3 \, dx = -8 \times \frac{1}{4} = -2\]
By following these calculations, we solve each integral separately and combine them to find the overall definite integral value, offering insight into the net area covered.
Step-by-step Integration Process
Solving integrals systematically can be greatly simplified by breaking the task into smaller parts. This involves:
  • Setting up the integrals: Write the integral of the polynomial by separating each term.
  • Calculating each integral: For polynomial terms, use basic integration rules, where the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\).
  • Evaluating each definite integral: Substitute the integration limits into the antiderivative and solve to get numerical results.
  • Combining results: Add or subtract the results of each integral calculation to find the net value.
By carefully executing each step, you enhance your understanding and ensure accuracy in calculating definite integrals.

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Most popular questions from this chapter

Evaluate the Riemann sums \(L_{4}\) and \(R_{4}\) for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. $$y=\sqrt{x}+\frac{1}{x} \text { over }[1,4]$$

Estimate \(\int_{0}^{1} t d t\) by comparison with the area of a single rectangle with height equal to the value of \(t\) at the midpoint \(t=\frac{1}{2} .\) How does this midpoint estimate compare with the actual value \(\int_{0}^{1} t d t ?\)

In the following exercises, use the evaluation theorem to express the integral as a function \(F(x)\) . $$ \int_{1}^{x} e^{t} d t $$

Estimate \(\int_{0}^{1} t d t\) using the left and right endpoint sums, each with a single rectangle. How does the average of these left and right endpoint sums compare with the actual value \(\int_{0}^{1} t d t ?\)

In the following exercises, compute the average value using the left Riemann sums \(L_{N}\) for \(N=1,10,100\) . How does the accuracy compare with the given exact value? Suppose that \( A=\int_{-\pi / 4}^{\pi / 4} \sec ^{2} t d t=\pi \qquad\) and \(B=\int_{-\pi / 4}^{\pi / 4} \tan ^{2} t d t .\) Show that \(A-B=\frac{\pi}{2}\).
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