Problem 36 - Calculus Early Transcendentals

Question

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\ln 8} e^{x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral $$\int_{0}^{\ln{8}} e^{x} dx$$ using the Fundamental Theorem of Calculus. Answer: The value of the integral is 7.

Step 1: Find the antiderivative of the integrand

The antiderivative of e^x is e^x itself, since the derivative of e^x is e^x. Therefore, the antiderivative (F(x)) is e^x.

Step 2: Use the Fundamental Theorem of Calculus to compute the definite integral

Now that we have the antiderivative, F(x) = e^x, we can use the theorem to compute the integral: $$\int_{0}^{\ln{8}} e^{x} dx = F(\ln{8}) - F(0)$$

Step 3: Evaluate F(ln(8)) and F(0)

We have F(x) = e^x. Thus, we evaluate F at the endpoints: F(ln(8)) = e^{(\ln{8})} F(0) = e^0

Step 4: Simplify the results

Using properties of exponents and logarithms, we can simplify: F(ln(8)) = e^{(\ln{8})} = 8 F(0) = e^0 = 1

Step 5: Find the value of the integral

We can now compute the integral using our simplified F values: $$\int_{0}^{\ln{8}} e^{x} dx = F(\ln{8}) - F(0) = 8 - 1 = 7$$ Hence, the value of the integral is 7.

Key Concepts

Antiderivative

In calculus, an antiderivative is a function whose derivative yields the original function. Think of it as "reversing" the process of differentiation. If you take the derivative of the antiderivative, you end up back at the original function.
For exponential functions like \( e^x \), finding the antiderivative can be straightforward because \( e^x \) is its own antiderivative. This means that the integral \( \int e^x \, dx \) is simply \( e^x + C \), where \( C \) is the constant of integration.

• The process involves reversing differentiation.
• Exponential functions frequently have simple antiderivatives.
• Understanding derivatives and antiderivatives reciprocal relationship helps in mastering calculus.

When solving integrals, you'll often find the antiderivative first, which is then evaluated at specific limits using the Fundamental Theorem of Calculus.

Definite Integral

A definite integral computes the accumulation of values of a function across an interval. It's represented as \( \int_a^b f(x) \, dx \). Here, "\( a \)" is the starting point and "\( b \)" is the endpoint of the interval over which the function \( f(x) \) is integrated. Unlike an indefinite integral, which includes a constant \( C \), a definite integral results in a numerical value.
To evaluate a definite integral, the Fundamental Theorem of Calculus comes into play. This powerful theorem states that if \( F(x) \) is an antiderivative of \( f(x) \) over an interval [a, b], then \( \int_a^b f(x) \, dx = F(b) - F(a) \).

• Evaluate the antiderivative at the upper and lower limits separately.
• Subtract the lower limit value from the upper limit value.
• Definite integrals represent the net area under a curve between two points.

In our example, finding the antiderivative \( F(x) = e^x \) and applying it to the limits \( 0 \) and \( \ln(8) \), we subtract to obtain \( 8 - 1 = 7 \). The definite integral evaluates to a concrete number, indicating accumulation or net area.

Exponential Functions

Exponential functions are characterized by a constant base raised to the power of a variable, commonly represented as \( e^x \). They possess unique properties, such as their derivatives and antiderivatives being the same. This characteristic makes calculations with exponential functions often more straightforward than with other types of functions.
The base \( e \) is a mathematical constant approximately equal to 2.71828, famously arising in various natural and mathematical contexts. It simplifies various calculus operations, such as differentiation and integration. When evaluating expressions like \( e^{(\ln(b))} \), it simplifies to \( b \) due to properties of logarithms.

• Exponential functions model growth and decay processes, from populations to radioactive decay.
• Key property: the derivative of \( e^x \) is \( e^x \).
• Useful in solving differential equations and integrals.

Thus, understanding exponential functions is vital to mastering topics like integration, as seen in our example where \( e^{(\ln(8))} \) simplifies beautifully to \( 8 \), showcasing the power and elegance of logarithmic and exponential properties together.