Problem 8 - 5 Steps to A 5: AP Calculus AB 2021

Question

Evaluate the following definite integrals. $$ \int_{\ln 2}^{\ln 3} 10 e^{x} d x $$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral of the function \(10e^x\) with respect to \(x\) over the interval \([\ln 2, \ln 3]\). Answer: The definite integral of \(10e^x\) over the interval \([\ln 2, \ln 3]\) is \(10\).

Step 1: Find the antiderivative of the given function

Given the function \(10e^x\), we seek an antiderivative, which is a function whose derivative is equal to the function. In other words, we want to find a function F(x) such that $$ F'(x) = 10e^x $$ The derivative of \(e^x\) is itself \(e^x\). Therefore, the antiderivative of \(10e^x\) with respect to x is \(10e^x\), so $$ F(x) = 10e^x $$

Step 2: Evaluate the antiderivative at the limits of integration

Now, according to the Fundamental Theorem of Calculus, we need to find the difference between the antiderivative evaluated at the upper limit of integration and the antiderivative evaluated at the lower limit of integration. So, $$ F(\ln 3) = 10e^{\ln 3} $$ and $$ F(\ln 2) = 10e^{\ln 2} $$ Recall that \(e^{\ln a} = a\), so we get $$ F(\ln 3) = 10 \cdot 3 = 30 $$ and $$ F(\ln 2) = 10 \cdot 2 = 20 $$

Step 3: Calculate the definite integral

Finally, subtract the antiderivative values, we get $$ \int_{\ln 2}^{\ln 3} 10 e^{x} d x = F(\ln 3) - F(\ln 2) = 30 - 20 = 10 $$ Thus, the definite integral of \(10e^x\) over the interval \([\ln 2, \ln 3]\) is \(10\).

Key Concepts

Antiderivative

An antiderivative is an essential idea in calculus. It involves finding a function whose derivative equals a given function. Consider the task of identifying an antiderivative for a simple exponential function like \(10e^x\). The antiderivative of \(e^x\) remains \(e^x\), so the antiderivative for \(10e^x\) turns into \(10e^x\) as well.
This is because when you differentiate \(10e^x\), the derivative is \(10e^x\), which matches the original function. Simply put, an antiderivative "works backward" from differentiation.

• It's important to remember that antiderivatives are not unique. They can differ by a constant.
• For example, if \(F(x) = 10e^x\) is an antiderivative, so is \(F(x) = 10e^x + C\), where \(C\) is any constant.

Finding antiderivatives is the first step in calculating definite integrals, which gives us the area under a curve between two points.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between differentiation and integration. It consists of two main parts:
The first part states that if you integrate a function and then differentiate it, you return to the original function.
The second part, which is more relevant for our exercise, asserts that the integral of a function over a closed interval \([a, b]\) can be computed using any one of its antiderivatives. Formally, it means:\[\int_a^b f(x) \, dx = F(b) - F(a)\]where \(F(x)\) is an antiderivative of \(f(x)\).

With our example, if \(f(x) = 10e^x\), the use of the theorem allows us to find:

• \(F(\ln 3) = 10 \, \cdot \, 3 = 30\)
• \(F(\ln 2) = 10 \, \cdot \, 2 = 20\)

Therefore, the definite integral calculation simplifies to: \(F(\ln 3) - F(\ln 2) = 30 - 20 = 10\).

This shows how integrating connects with finding areas under curves, connecting two core aspects of calculus.

Exponential Functions

Exponential functions are a key concept in mathematics and calculus. They take the form \(f(x) = a \cdot e^{bx}\), where \(e\) is the base of natural logarithms, approximately equal to 2.718.

Let's explore why they're special:

• The derivative of \(e^x\) is \(e^x\). This property is unique to exponential functions with base \(e\), making calculus operations on them straightforward.
• The function \(10e^x\) is simply a scaled version of \(e^x\). Each x value gets multiplied by 10, scaling the output.

Exponential functions widely appear in models of growth and decay, such as population growth or radioactive decay.
The ability to integrate these functions, as we've seen, is crucial for understanding a range of natural phenomena. In our exercise, the function \(10e^x\) integrated over a specific interval gave a tangible result, showing how exponential functions interact dynamically in calculus.