Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{1} 10 e^{x+3} d x$$

Short answer

Question: Evaluate the definite integral: $$\int_{0}^{1} 10 e^{x+3} d x$$ Answer: The definite integral evaluates to \(10(e^4-e^3)\).

Step-by-step solution

Find the antiderivative of the function

The given function is \(10 e^{x+3}\). To find the antiderivative, we need to find a function whose derivative is equal to the given function. The derivative of \(e^x\) is itself, so the derivative of \(e^{x+3}\) is \(e^{x+3}\). Thus, the antiderivative of \(10e^{x+3}\) is \(10e^{x+3} + C\) (where C is the constant of integration).

Apply the Fundamental Theorem of Calculus

Now that we found the antiderivative, we can use the FTC to evaluate the definite integral. The FTC states that if \(F(x)\) is an antiderivative of \(f(x)\), then: $$\int_{a}^{b} f(x) d x = F(b) - F(a)$$ In our case, \(F(x) = 10e^{x+3} + C\), \(a = 0\), and \(b = 1\). Applying the FTC, we have: $$\int_{0}^{1} 10 e^{x+3} d x = \left[10e^{x+3}\right]_0^1 = 10e^4-10e^3$$

Simplify the expression

Now, we just need to simplify the expression: $$10e^4-10e^3 = 10(e^4-e^3)$$ The definite integral $$\int_{0}^{1} 10 e^{x+3} d x$$ evaluates to \(10(e^4-e^3)\).

Key concepts

Definite Integrals

Definite integrals are an essential tool in calculus and help calculate the net area under a curve. They are often used to find total accumulation of quantities, such as area, volume, or any other summation where the limits define a range. In our problem, we have the definite integral from 0 to 1 of the function.

• The integral sign is followed by limits of integration, indicating boundaries: 0 is the lower limit, and 1 is the upper limit.
• The function being integrated, here \(10 e^{x+3}\), is called the integrand.
• The differential \(dx\) signifies the variable of integration.

To solve the definite integral using the Fundamental Theorem of Calculus, we must first find the antiderivative of the integrand and then evaluate at the upper and lower limits. This process not only showcases the power and efficiency of definite integrals but also crucially connects differentiation and integration.

Antiderivative

An antiderivative, sometimes referred to as an indefinite integral, is a function whose derivative equals the original function. In simple terms, finding an antiderivative means reversing the process of differentiation.For the given problem, we need the antiderivative of \(10 e^{x+3}\). Let's break down the steps:

• The exponential function \(e^{x}\) has a unique property where its derivative and antiderivative are the same, \(e^{x}\).
• When constants are involved, like the 10 in our function, they factor out and multiply the result.
• Thus, the antiderivative of \(10 e^{x+3}\) is \(10 e^{x+3} + C\). Here, \(C\) represents the constant of integration, which is crucial for indefinite integrals, but disappears in definite integrals when limits are applied.

Understanding how to find antiderivatives is pivotal as they form the backbone for evaluating definite integrals and thus deepen our understanding of accumulation processes in calculus.

Exponential Functions

Exponential functions, characterized by a constant base raised to a variable exponent, play a significant role in calculus. They show up in many natural phenomena due to their self-replicating property.For example, \(e^{x}\) is the natural exponential function, where \(e\) is Euler's number, approximately equal to 2.718. It has unique characteristics that make it particularly useful:

• The rate of growth of \(e^{x}\) is proportional to its current value, an attribute that appears in processes such as compound interest, population growth, and radioactive decay.
• Differentiation and integration operations are quite straightforward: \(\frac{d}{dx} e^{x} = e^{x}\), and its antiderivative is also \(e^{x}\).
• When it comes to functions like \(e^{x+3}\), the expression \(x+3\) shifts the curve horizontally, but the essential properties remain intact.

Mastering exponential functions simplifies working with derivatives and integrals in calculus, making them a powerful tool when handling growth models and real-world data.