Problem 31 - Applied Calculus

Question

Use the Fundamental Theorem to determine the value of \(b\) if the area under the graph of \(f(x)=4 x\) between \(x=1\) and \(x=b\) is equal to \(240 .\) Assume \(b>1\).

Step-by-Step Solution

Answer

\( b = 11 \).

Step 1: Understand the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if a function \( f \) is continuous over \([a, b]\), then the area under the curve of \( f(x) \) between \( a \) and \( b \) is equal to the definite integral \( \int_{a}^{b} f(x) \, dx \). Here, \( a = 1 \), \( b = ? \), and \( f(x) = 4x \). We want this integral to be 240.

Step 2: Set up the Definite Integral

Use the expression for the definite integral: \( \int_{1}^{b} 4x \, dx \). We know this integral equals 240, so we set the equation: \( \int_{1}^{b} 4x \, dx = 240 \).

Step 3: Find the Antiderivative

The antiderivative of \( 4x \) is \( 2x^2 \) because \( \frac{d}{dx} (2x^2) = 4x \).

Step 4: Evaluate the Definite Integral

Apply the limits of integration to the antiderivative: \[ 2(b)^2 - 2(1)^2 = 240 \]. This simplifies to \[ 2b^2 - 2 = 240 \].

Step 5: Solve for \( b \)

Rearrange the equation \( 2b^2 - 2 = 240 \) to solve for \( b \):\[ 2b^2 = 242 \] \[ b^2 = 121 \] \[ b = \sqrt{121} \]. Since \( b > 1 \), \( b = 11 \).

Key Concepts

Definite Integrals

When we talk about definite integrals, we are referring to a way to find the total area under a curve that is defined by a continuous function. This process is crucial in calculus as it provides a method to measure the accumulation of quantities, such as areas, volumes, and other accumulated change.To compute a definite integral, we take two values from the domain of a function, called the limits of integration. These are usually denoted as \(a\) and \(b\), where \(a < b\). The definite integral of a function \(f(x)\) from \(a\) to \(b\) is noted as \( \int_{a}^{b} f(x) \, dx \). Essentially, this expression calculates the net area between the \(x\)-axis and the graph of \(f(x)\).

• The integral sign \(\int\) signifies integration.
• \(f(x)\) represents the function we want to integrate.
• \(dx\) indicates the variable of integration, which, in most basic cases, is \(x\).
• The limits \(a\) and \(b\) are the bounds of the interval we are considering for the area calculation.

In our specific exercise, we are tasked with finding the area under the curve of \(f(x) = 4x\) from 1 to some unknown value \(b\), which equals 240. By setting the definite integral to equal 240, we can solve for \(b\) to locate the desired point.

Antiderivatives

An antiderivative, sometimes known as an indefinite integral, is the reverse operation of taking a derivative. If you remember derivatives as the process that gives us the rate of change of a function, then the antiderivative returns us to the original function, up to a constant. Consider a function \(f(x)\). Its antiderivative is another function \(F(x)\) such that when you differentiate \(F(x)\), you get \(f(x)\). Mathematically, this is expressed as \(F'(x) = f(x)\).

• An antiderivative of \(f(x) = 4x\) is \(2x^2\), since \(\frac{d}{dx}(2x^2) = 4x\).
• We can determine antiderivatives by reversing known derivative rules or using integration techniques.
• While a function can have many antiderivatives (differing only by a constant), they are central to solving integrals, especially when evaluating definite integrals.

In our example, finding the antiderivative was vital to solving the definite integral \(\int_{1}^{b} 4x \, dx\). Knowing the antiderivative, \(2x^2\), allowed us to apply the limits and solve for \(b\) so that the given area condition was satisfied.

Continuous Functions

Continuous functions are essential in the context of integration. A function is considered continuous over an interval if there are no breaks, jumps, or holes within that interval. In other words, you could draw the graph of the function on that interval without lifting your pen off the paper.For the Fundamental Theorem of Calculus to apply, the function we're working with must be continuous over the interval defined by the limits of integration: from \(a\) to \(b\).

• A continuous function does not "jump" or have abrupt changes within the specified interval.
• Continuity ensures that our integral truly represents the total accumulation (or area) up to that point.
• In our problem, \(f(x) = 4x\) is continuous over all real numbers, especially from \(x=1\) to \(x=b\).

Because \(f(x) = 4x\) is continuous everywhere, we were able to directly apply integral calculus to find the value of \(b\) using the definite integral, as the Fundamental Theorem of Calculus guarantees the existence and value of that integral.