Problem 342 - Calculus Volume 1

Question

Find the area under y = 1/x and above the x-axis from x = 1 to x = 4.

Step-by-Step Solution

Answer

The area is \( \ln(4) \).

Step 1: Understand the Problem

To find the area under the curve of the function \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 4 \) and above the x-axis. This is equivalent to finding the definite integral of \( y = \frac{1}{x} \) over this interval.

Step 2: Set Up the Definite Integral

The problem asks for the area under the curve from \( x = 1 \) to \( x = 4 \). The corresponding definite integral is \( \int_1^4 \frac{1}{x} \, dx \).

Step 3: Integrate the Function

To solve the integral \( \int \frac{1}{x} \, dx \), recall that the antiderivative of \( \frac{1}{x} \) is \( \ln|x| \). Therefore, the integral becomes \( \int_1^4 \frac{1}{x} \, dx = [\ln |x|]_1^4 \).

Step 4: Evaluate the Integral using the Fundamental Theorem of Calculus

We apply the fundamental theorem of calculus, which tells us to evaluate the antiderivative at the upper limit and subtract the value of the antiderivative at the lower limit: \( [\ln |x|]_1^4 = \ln|4| - \ln|1| \).

Step 5: Simplify the Expression

Having \( \ln|4| - \ln|1| = \ln(4) - 0 \), because \( \ln(1) = 0 \). So, the simplified result is \( \ln(4) \).

Key Concepts

Area Under a Curve

The area under a curve represents the space between a function and the x-axis over a specified range of x-values. It is a fundamental concept in calculus often computed using definite integrals. In our example, we are calculating the area under the curve defined by \( y = \frac{1}{x} \), over the interval from \( x = 1 \) to \( x = 4 \). Calculating areas in this manner allows us to understand how a quantity accumulates over time, space, or another variable.

• The x-values give the interval over which the area is evaluated.
• The y-values, being the function \( \frac{1}{x} \), show how the height of the curve changes.
• Integrals help sum an infinite number of infinitely small areas between the curve and the x-axis.

Understanding the area under the curve is crucial for interpreting real-world phenomena modeled by graphs, such as physics' speed-time graphs or economics' cost-revenue models. Although visualizing this area graphically can sometimes suffice, definite integrals give precise numerical values for these areas.

Fundamental Theorem of Calculus

The fundamental theorem of calculus bridges the concepts of derivative with the definite integral. There are two parts to this theorem: it shows that an antiderivative can be found using indefinite integration and connects this to evaluating definite integrals.
For the integral \( \int_1^4 \frac{1}{x} \, dx \), we applied the fundamental theorem of calculus to evaluate the antiderivative \( \ln|x| \) at specified bounds.

• This method involves plugging the upper limit (4) and lower limit (1) of the integral into the antiderivative.
• It then computes \( \ln(4) - \ln(1) \).
• Since \( \ln(1) = 0 \), the final result simplifies to \( \ln(4) \).

The elegance of this theorem is in its simplicity. It utilizes an indefinite integral (antiderivative) to find the accumulated value over an interval. This crucial connection elevates our understanding of calculus as a whole and enhances our problem-solving in areas of rate changes and quantities accumulation.

Antiderivative

An antiderivative of a function is essentially the reverse of differentiation. If you differentiate an antiderivative, you should return to the original function. In this example, the function \( y = \frac{1}{x} \) has the antiderivative \( \ln|x| \).

• The antiderivative represents a family of functions that vary by a constant, usually represented as \( C \).
• Antiderivatives help calculate areas, positions, and accumulations where integration is involved.
• Understanding the reverse process of differentiation broadens how we can work with functions.

By finding the antiderivative, we can evaluate definite integrals and solve problems related to total change over specified intervals. This process enables precise calculations of areas and totals, crucial for accurate mathematical modeling across disciplines.