Problem 101 - Calculus Volume 1

Question

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. \(\int_{0}^{1}(1-2 x)^{3} d x\)

Step-by-Step Solution

Answer

The integral is 0.

Step 1: 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:\[(1 - 2x)^3 = 1 - 6x + 12x^2 - 8x^3\]

Step 2: Set Up Integrals

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

Step 3: 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. \(- 8 \int_{0}^{1} x^3 \, dx = -8 \times \frac{1}{4} = -2\)

Step 4: Combine the Results

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

Step 5: Conclusion

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

Key Concepts

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.