Problem 68 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int_{0}^{1} \frac{16^{x}}{4^{2 x}} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral of the function $\frac{16^x}{4^{2x}}$ from x = 0 to x = 1. Answer: The definite integral is equal to 1.

Step 1: Simplify the integrand

We can simplify the integrand by recognizing that both terms in the numerator and denominator have the same base power, which is \(4\). We rewrite the integrand as follows: $$\frac{16^x}{4^{2x}} = \frac{(4^2)^x}{4^{2x}} = \frac{4^{2x}}{4^{2x}} = 1$$ Now our integral simplifies to: $$\int_{0}^{1} 1 \, dx$$

Step 2: Find the antiderivative of the simplified function

The antiderivative of the constant function \(f(x) = 1\) is simply \(F(x) = x + C\), where \(C\) is the constant of integration. Since we are dealing with a definite integral, we don't need to worry about the integration constant.

Step 3: Evaluate the antiderivative at the limits

Next, we need to evaluate the antiderivative at the given limits. First, find \(F(1)\): $$F(1) = 1$$ Now, find \(F(0)\): $$F(0) = 0$$

Step 4: Calculate the definite integral

Finally, subtract the value of the antiderivative at the lower limit from the value at the upper limit to get the definite integral: $$\int_{0}^{1} 1 \, dx = F(1) - F(0) = 1 - 0 = 1$$ So, the definite integral is equal to 1.

Key Concepts

Integration Techniques

In the realm of calculus, integration techniques play a vital role in solving complex problems involving areas under curves, among other applications. The exercise provided requires us to evaluate a definite integral involving an exponential function. The key steps involve simplifying the integrand before integrating. By identifying common factors, like exponent bases, and rewriting the function in its simplest form, we can streamline the integration process.
For this particular problem, simplification is achieved by using the fact that both terms in the integrand share the same base power, leading to simplification to a constant function.

• Recognize common bases: This helps simplify expressions for easier integration.
• Simplify the integrand: Reduces the complexity of the integral, often to a basic function, as shown in our exercise.
• Evaluate the definite integral: Involves finding the antiderivative and assessing it over specified limits, simplifying the process of area calculation.

Mastering these integration techniques enhances the ability to tackle a wide range of problems efficiently by minimizing algebraic complexity before performing the actual integration.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions appear frequently in calculus, especially within integration problems. In the given exercise, the function involves exponential terms with a common base of 4.
Understanding how to manipulate these bases is crucial for solving integrals.

• Identifying the common base: Both numerator and denominator have a base of 4, which can be simplified through known exponent rules.
• Manipulating exponents: Employ rules such as product or power of exponentials to reduce complexity.
• Simplification to a basic form: Recognize opportunities for reducing an expression, as demonstrated by simplifying \( \frac{16^x}{4^{2x}} \) to 1.

By comprehending these exponential functions, students can simplify integrals and perform further calculus operations with ease. This understanding is pivotal when approaching integrations involving complex exponential relationships.

Antiderivatives

Antiderivatives, also known as indefinite integrals, are functions that represent the accumulation of a quantity, reversing the process of differentiation. In the context of definite integrals, antiderivatives help evaluate the total accumulation between two limits.
Once our original integrand is reduced to a constant function as in this problem, the antiderivative becomes straightforward. The antiderivative of a constant function like 1 is a linear function of the form \(x + C\), where \(C\) is the integration constant. However, in definite integrals, this constant does not affect the final result.

• Find the antiderivative: Determine the function whose derivative is the original function, which for constants is simple and direct.
• Evaluate at specified limits: Apply the fundamental theorem of calculus to find the net value between two domain points, as seen in evaluating \(F(x)\) between 0 and 1.
• Subtract evaluations: Calculate the difference, simplifying the evaluation of the definite integral.

Comprehending antiderivatives enables the execution of definite integrals with greater clarity and precision, reinforcing the inverse relationship between differentiation and integration.