Problem 60 - Applied Calculus

Question

Evaluate each definite integral. $$ \int_{0}^{2} 3 e^{x / 2} d x $$

Step-by-Step Solution

Answer

The value of the integral \( \int_{0}^{2} 3 e^{x/2} \, dx \) is \( 6e - 6 \).

Step 1: Understand the problem

We are asked to evaluate the definite integral \( \int_{0}^{2} 3 e^{x/2} \, dx \). This involves finding the integral of the function from 0 to 2.

Step 2: Identify the antiderivative

The integrand is the exponential function \( 3 e^{x/2} \). To find the antiderivative, recall that the derivative of \( e^{ax} \) is \( a e^{ax} \). Therefore, the antiderivative of \( 3 e^{x/2} \) can be found by dividing by the derivative of \( x/2 \), which is 1/2.

Step 3: Find the antiderivative

The antiderivative of \( 3 e^{x/2} \) is computed as follows: divide the constant 3 by the derivative of \( x/2 \), which is 1/2. This gives us:\[ 6 e^{x/2} \].

Step 4: Apply the limits of integration

Use the fundamental theorem of calculus to evaluate the definite integral. Substitute the upper limit 2 and the lower limit 0 into the antiderivative:\[ 6 e^{x/2} \bigg|_0^2 = 6 e^{2/2} - 6 e^{0/2} \].

Step 5: Simplify the expression

Compute the exponents:\[ 6 e^{1} - 6 e^{0} = 6e - 6(1) \].

Step 6: Calculate the final result

Simplify the expression to get the final result:\[ 6e - 6 \].

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the world of integrals and derivatives. It consists of two main parts, and plays a crucial role in evaluating definite integrals.
The first part tells us that if we have a continuous function, the antiderivative of this function will provide the net area under the curve from one point to another. Essentially, it guarantees that every continuous function has an antiderivative.
The second part, which is often more directly used, states that if you have an antiderivative for a function, you can find the definite integral of that function over a certain interval by subtracting the values of the antiderivative at the boundaries of the interval.

• For our exercise, once we found the antiderivative of the function, which was \(6e^{x/2}\), we applied the fundamental theorem by substituting in the upper and lower limits, 2 and 0, into the antiderivative.
• This allowed us to evaluate the definite integral by calculating \(6 e^{2/2} - 6 e^{0/2}\).

This process emphasizes how pivotal the fundamental theorem of calculus is in simplifying the evaluation of definite integrals.

Exponential Function Integration

Integrating exponential functions can initially seem tricky, but they follow a straightforward pattern.
The exponential function \(e^x\) is unique because it is its own derivative; however, when the exponent is more complex, such as \(e^{ax}\), the integral takes a different form.

• In general, the integral of \(e^{ax}\) is \(\frac{1}{a}e^{ax}\), found by reversing the process of differentiation, essentially undoing the multiplication of the exponent's coefficient.
• In our exercise, we dealt with the function \(3e^{x/2}\). To find the integral, we divided the constant 3 by the derivative of \(x/2\), which is \(1/2\), yielding the antiderivative \(6e^{x/2}\).

Recognizing this pattern makes solving exponential integrals more manageable, allowing us to consistently apply this knowledge to similar problems.

Antiderivative

An antiderivative is essentially a "reverse" derivative. It is a function whose derivative returns the original function.
Finding the antiderivative is key to solving integrals, especially definite integrals, because it allows us to apply the Fundamental Theorem of Calculus.

• To find the antiderivative of a function like \(3e^{x/2}\), you need to consider what function, when differentiated, gives you \(3e^{x/2}\).
• Because the derivative of \(e^{x/2}\) includes dividing by \(1/2\), finding its antiderivative involves multiplying by \(2\), resulting in \(6e^{x/2}\).

Understanding how to calculate antiderivatives is crucial, as it provides us with the tool needed to evaluate definite integrals.
This practice sets the stage for solving complex integrals in various mathematical and applied contexts.