Problem 31 - Calculus Single Variable

Question

In each of Exercises \(31-34,\) calculate the given definite integral. \(\int_{0}^{1} 2^{x} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{1}{\ln(2)}\).

Step 1: Identify the Function

The function we need to integrate is given by \(f(x) = 2^x\). The limits of integration are from 0 to 1.

Step 2: Find the Antiderivative

To find the antiderivative of \(2^x\), recall that the integral of \(a^x\) is \(\frac{a^x}{\ln(a)} + C\) for any constant \(a > 0, a eq 1\). Here, \(a = 2\), so the antiderivative of \(2^x\) is \(\frac{2^x}{\ln(2)}\).

Step 3: Use the Fundamental Theorem of Calculus

Apply the Fundamental Theorem of Calculus, which states: If \(F(x)\) is the antiderivative of \(f(x)\), then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\). We calculate \(F(1) - F(0)\) where \(F(x) = \frac{2^x}{\ln(2)}\).

Step 4: Evaluate the Antiderivative at the Limits

Compute \(F(1) = \frac{2^1}{\ln(2)} = \frac{2}{\ln(2)}\) and \(F(0) = \frac{2^0}{\ln(2)} = \frac{1}{\ln(2)}\).

Step 5: Calculate the Definite Integral

Subtract to find the definite integral: \(\int_{0}^{1} 2^x \, dx = F(1) - F(0) = \frac{2}{\ln(2)} - \frac{1}{\ln(2)} = \frac{2 - 1}{\ln(2)} = \frac{1}{\ln(2)}\).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a cornerstone concept in the field of calculus. It connects two primary branches: differentiation and integration. This theorem states that if you have a continuous function, the process of taking the derivative and the process of integrating are inverse operations. There are two parts to this theorem:

• First Part: It tells us that if a function is continuous over a certain interval, then it has an antiderivative.
• Second Part: This is the one we often use for calculations. It states that if you can find an antiderivative of a function, you can use it to compute the definite integral. You do this by evaluating the antiderivative at the upper limit and subtracting the value of the antiderivative at the lower limit.

In this exercise, the function we have is an exponential one, specifically, the function is given as \( f(x) = 2^x \). By finding its antiderivative and using the theorem, we can easily compute the definite integral \( \int_{0}^{1} 2^x \, dx \). This allows us to find the exact area under the curve of \( f(x) = 2^x \) from 0 to 1.

Antiderivative

The antiderivative of a function is also known as the indefinite integral. It's essentially the reverse process of differentiation, where for a given function \(f(x)\), we find another function \(F(x)\) such that the derivative \(F'(x) = f(x)\). The challenge is knowing the integration formulas and applying them correctly.In our case, we're dealing with \( 2^x \), which is an exponential function. The general rule for finding the antiderivative of an exponential function \(a^x\) (where \(a > 0\) and \(a eq 1\)) is given by:\[\int a^x \, dx = \frac{a^x}{\ln(a)} + C\]Here, \(C\) is the constant of integration. Applying this formula to \( 2^x \), we get the antiderivative:\[F(x) = \frac{2^x}{\ln(2)}\] This result is fundamental when calculating the definite integral using the Fundamental Theorem of Calculus. It allows us to determine the precise area under the curve of the function within specified limits.

Exponential Function

Exponential functions are a crucial class of functions in mathematics, characterized by the input variable in the exponent. They are often represented as \( f(x) = a^x \), where \(a\) is a positive constant. These functions are notable for their unique properties:

• They always have an increasing (or in cases where \(0 < a < 1\), decreasing) behavior as \(x\) increases.
• The functions never touch the x-axis, meaning \(a^x\) is never zero, and they have a horizontal asymptote at y=0.
• They have an important property where the rate of growth (or decay) is proportional to their current value, making them ideal for modeling certain real-world situations, such as population growth or radioactive decay.

In this exercise, we are looking at the function \(f(x) = 2^x\). Its antiderivative, which was calculated using the rule for exponential functions, enables us to evaluate integrals easily. This is particularly useful in calculus, as it helps in finding areas, averages, and solutions to growth and decay problems.