Problem 38 - Calculus Single Variable

Question

Evaluate the given definite integral by finding an antiderivative of the integrand and applying Theorem \(3 .\) $$ \int_{-1}^{0} e^{1+x} d x $$

Step-by-Step Solution

Answer

The value of the integral is \( e - 1 \).

Step 1: Identify the Integrand

The integrand given in the exercise is \( e^{1+x} \). We will be finding an antiderivative for this function, which is needed to apply the Fundamental Theorem of Calculus (Theorem 3).

Step 2: Find the Antiderivative

To find the antiderivative of \( e^{1+x} \), recognize that the derivative of \( e^{1+x} \) with respect to \(x\) is \( e^{1+x} \). Integrating \( e^{1+x} \) results in \( e^{1+x} + C \), where \(C\) is the constant of integration.

Step 3: Apply the Limits

Now, apply the limits of integration from \(-1\) to \(0\) in the antiderivative found in Step 2. This gives us \( \left[e^{1+x} \right]_{-1}^{0} \).

Step 4: Evaluate the Antiderivative at the Limits

Evaluate \( e^{1+x} \) at the upper limit (\(x = 0\)) and subtract its value at the lower limit (\(x = -1\)). Thus, we compute \( e^{1+0} - e^{1+(-1)} = e - e^{0} = e - 1 \).

Step 5: Conclusion

The value of the definite integral \( \int_{-1}^{0} e^{1+x} \ dx \) is found to be \( e - 1 \).

Key Concepts

Antiderivative

When we talk about an antiderivative, we are looking for a function whose derivative gives us the original function. In this exercise, we start with the function \( e^{1+x} \) and need to identify an antiderivative. Recognizing the exponential function simplifies the task - the antiderivative is again \( e^{1+x} \) plus a constant, though we don't need to keep the constant for definite integrals. Finding an antiderivative is the reverse process of differentiation.

• Differentiation: From a function to its rate of change.
• Antidifferentiation: From a rate of change back to the function.

Remember, the antiderivative given by \( e^{1+x} + C \) represents a whole family of functions, but in definite integrals, the constants cancel out when applying limits.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration. It's the bridge that makes evaluating definite integrals possible. For the exercise, this theorem facilitates the process in which you first determine an antiderivative of the integrand. Once you have it, you can straightforwardly determine the value of the definite integral

• Find the antiderivative of the integrand.
• Evaluate this antiderivative between the given limits.

The theorem signifies that the area under the curve of a function over an interval \([a,b]\) can be computed using the difference between the antiderivative values at these bounds. It effectively turns a challenging estimation problem into a straightforward calculation.

Limits of Integration

In the context of definite integrals, limits of integration define where you start and stop calculating the area under the curve. For this exercise, the limits are from \(-1\) to \(0\). These limits are applied to the antiderivative function to accurately compute the net area. To use these limits:

• Evaluate the antiderivative at the upper limit.
• Evaluate the antiderivative at the lower limit.
• Subtract the lower limit result from the upper limit result.

In practice, this means evaluating \( e^{1+x} \) at \(x = 0\) and \(x = -1\), calculating \(e - 1 \), which provides the definite integral's numerical value.