Problem 191 - Calculus Volume 1

Question

In the following exercises, use the evaluation theorem to express the integral as a function \(F(x)\) . $$ \int_{1}^{x} e^{t} d t $$

Step-by-Step Solution

Answer

F(x) = e^{x} - e

Step 1: Identify the integral

We are given the integral \( \int_{1}^{x} e^{t} \, dt \). Our task is to express this integral as a function \( F(x) \).

Step 2: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \( f \) is continuous on \([a, b]\) and \( F \) is an antiderivative of \( f \) on \([a, b]\), then \( \int_{a}^{b} f(t) \, dt = F(b) - F(a) \). Here, \( f(t) = e^{t} \). An antiderivative of \( e^{t} \) is also \( e^{t} \).

Step 3: Compute the antiderivative

Find an antiderivative of \( e^{t} \), which is \( F(t) = e^{t} \).

Step 4: Evaluate the antiderivative at the limits of integration

Substitute into the Fundamental Theorem of Calculus: \( F(x) = e^{x} \) and \( F(1) = e^{1} \).

Step 5: Express the integral as a function

Using the results from Step 4, the integral becomes \( F(x) = e^{x} - e \). Hence, the function expressing the integral is \( F(x) = e^{x} - e \).

Key Concepts

Evaluation Theorem

The evaluation theorem is a critical component of the Fundamental Theorem of Calculus. It bridges the concept of an antiderivative with the evaluation of a definite integral. When given an integral such as \( \int_{1}^{x} e^{t} \, dt \), we use this theorem to express the integral as a function. The steps involve:

• Finding an antiderivative of the integrand, \( e^{t} \), which is also \( e^{t} \).
• Calculating the antiderivative at the limits, here at \( x \) and 1.
• The result becomes the difference \( F(x) - F(1) \), simplifying to the function \( F(x) = e^{x} - e \).

This provides a straightforward pathway to evaluate definite integrals and connect them with their antiderivatives, enabling further exploration of continuous functions and their behaviors.

Antiderivative

An antiderivative is a function whose derivative is the original function you started with. For an exponential function like \( e^{t} \), finding the antiderivative is simple because the derivative of \( e^{t} \) remains \( e^{t} \). This unique property makes it beneficial in calculus, particularly when applying the Fundamental Theorem of Calculus.
The antiderivative helps solve definite integrals by providing the necessary function to evaluate at the limits. In our exercise, finding \( F(t) = e^{t} \) allowed us to express the integral as a function \( F(x) = e^{x} - e \).
Keep in mind:

• Different functions have different antiderivatives.
• Antiderivatives often involve constants of integration when dealing with indefinite integrals.
• In definite integrals, constants cancel out during the evaluation.

Definite Integral

Definite integrals are the bread and butter of calculus, allowing us to compute the accumulation of quantities. When we consider \( \int_{1}^{x} e^{t} \, dt \), we are working with a definite integral between two limits: from 1 to \( x \).
This integral evaluates how the function \( e^{t} \) acts over an interval, quantifying the total 'area' under the curve from 1 to \( x \).
Important aspects:

• Definite integrals yield a numerical value representing the net signed area.
• They use limits of integration to determine start and end points.
• The Fundamental Theorem of Calculus makes them feasible to calculate using antiderivatives.

The evaluation of the definite integral \( \int_{1}^{x} e^{t} \, dt \) illustrates how calculus translates functions into quantitative results.

Continuous Function

A continuous function smoothly connects its values without breaks, jumps, or holes. It's a crucial property required for applying the Fundamental Theorem of Calculus, as seen in the function \( f(t) = e^{t} \).
Continuous functions like \( e^{t} \) ensure reliable results when computing both indefinite and definite integrals. This reliability stems from guaranteed well-behaved limits and incremental changes over intervals.
Features of continuous functions include:

• They are differentiable within their intervals of definition, strengthening their role in integration.
• The absence of discontinuities ensures seamless antiderivative finding and integration.
• Understanding continuity is vital for analyzing how a function changes and its corresponding graphical representation.

In the context of this exercise, recognizing \( e^{t} \) as continuous is fundamental for confidently applying calculus principles.