Problem 48 - Calculus Early Transcendentals

Question

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{1} 10 e^{2 x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral \(\int_{0}^{1} 10e^{2x} dx\). Answer: The definite integral is approximately 36.95.

Step 1: Find the antiderivative of the function

The given function is \(f(x) = 10e^{2x}\). To find the antiderivative of this function, we will use the power rule and chain rule for integration. Let \(F(x)\) represent the antiderivative, then: $$F(x) = \int 10 e^{2x} dx$$ Let \(u = 2x\). Then, \(du = 2dx\). So, \(dx = \frac{1}{2}du\). Now, substitute \(u\) in the expression: $$F(x) = \int 10 e^u \cdot \frac{1}{2} du$$ Now, integrate with respect to \(u\): $$F(x) = 5\int e^u du$$ The integral of \(e^u\) is itself: $$F(x) = 5e^u + C$$ Now, revert the substitution to get the antiderivative in terms of \(x\): $$F(x) = 5e^{2x} + C$$

Step 2: Apply the Fundamental Theorem of Calculus

Now that we have found the antiderivative, we can use the Fundamental Theorem of Calculus to evaluate the integral over the given interval \([0, 1]\). The theorem states: $$\int_{a}^{b} f(x) dx = F(b) - F(a)$$ For our exercise, we have the limits \(a = 0\) and \(b = 1\). Plug these values into the theorem: $$\int_{0}^{1} 10 e^{2x} dx = F(1) - F(0)$$ Now, substitute the antiderivative function \(F(x)\): $$\int_{0}^{1} 10 e^{2x} dx = (5e^{2(1)} + C) - (5e^{2(0)} + C)$$ Since the constant \(C\) appears in both terms, it will cancel out: $$\int_{0}^{1} 10 e^{2x} dx = 5e^{2} - 5e^{0}$$ Finally, evaluate the expression: $$\int_{0}^{1} 10 e^{2x} dx = 5e^{2} - 5 \approx 5(7.39) - 5 = 36.95$$ The definite integral is approximately 36.95.

Key Concepts

Definite Integral

Understanding the definite integral is crucial for solving problems in calculus. It represents the accumulated sum of function values between two points, a and b, on a graph. When you see an expression like \[ \int_{a}^{b} f(x) dx \], it's telling you to find the total area under the curve of function f(x) from point a to point b.

To compute a definite integral, you need two main components: the function you wish to integrate, and the limits of integration, which are the values a and b. In practice, this process converts a function into accumulated quantities, such as distance traveled when given a velocity function over time.

• First, find the antiderivative of the function, F(x).
• Then, apply the limits as F(b) - F(a), according to the Fundamental Theorem of Calculus.

For the exercise \[ \int_{0}^{1} 10 e^{2 x} dx \], where f(x) = 10e^{2x}, the area under the curve from 0 to 1 is computed following these steps.

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. In other words, if F(x) is the antiderivative of f(x), then F'(x) equals f(x). The process of finding an antiderivative is known as antidifferentiation or integration.

Each function can have many antiderivatives, differing by a constant, since the derivative of a constant is zero. That's why antiderivatives are often expressed with a '+ C', accounting for this constant of integration.

• To find an antiderivative, use integration techniques that reverse differential rules.
• For example, the antiderivative of 10e^{2x}, found in the provided exercise, involves recognizing e^u as its own antiderivative.

In practice, the antiderivative plays a vital role in the calculation of the definite integral by using the Fundamental Theorem of Calculus.

Integration Techniques

Several techniques can be used to solve integration problems, and choosing the right one depends on the form of the function you're working with. Common techniques include substitution, integration by parts, partial fraction decomposition, and trigonometric integration.

In the given exercise, we used substitution by letting u equal 2x. Substitution is useful when a function contains a composite function that would be simpler to integrate on its own — in this case, simplifying the integration of 10e^{2x} down to 5e^u.

• Substitute a section of the original function with a single variable to simplify.
• Execute the integration on this simpler form.
• Then, revert back to the original variable.

Such strategies reduce complex integrals to more manageable forms, widening the scope of functions that can be integrated.

Exponential Functions

Exponential functions are a class of mathematical functions of the form f(x) = a^x, where the base a is a constant. These functions show up frequently in various domains of science and mathematics due to their properties of growth and decay.

In calculus, the base e exponential function, f(x) = e^x, is especially important because it is its own derivative and antiderivative. This simplifies many problems involving growth and decay, as well as compound interest calculations in finance.

• The function e^x has the unique property that its rate of increase is proportional to its current value.
• When integrating, e^x (as well as its variants like e^{2x}) will often result in itself, possibly with a multiplication factor.

Functions that model exponential growth or decay make integration, such as the one in the exercise, neat and potent in applications spanning from economics to physics.