Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\), using the fundamental theorem of calculus. Then verify your result using geometry. $$f(x)=2 ;[1,4]$$

Step-by-Step Solution

Answer

The area under the graph of the function \(f(x) = 2\) on the interval \([1, 4]\) can be found using the fundamental theorem of calculus. An antiderivative of \(f(x) = 2\) is \(F(x) = 2x + C\), and applying the theorem, we get: \[ \int_{1}^{4} 2 \, dx = F(4) - F(1) = 6 .\] The area is also confirmed to be 6 using geometry since the graph forms a rectangle with a base of 3 and height of 2: \[ \text{Area} = \text{Base} \times \text{Height} = 3 \times 2 = 6 .\]

Step 1: Apply the Fundamental Theorem of Calculus

\(\int_1^4 2 \, dx = [2x]_1^4 = 2(4) - 2(1) = 8 - 2 = 6\)

Step 2: Verify using geometry

The graph of \(f(x) = 2\) is a horizontal line at height 2. The region from \(x = 1\) to \(x = 4\) is a rectangle with width \(4 - 1 = 3\) and height \(2\). Area = \(3 \times 2 = 6\). \(\checkmark\)

Key Concepts

Antiderivative

An antiderivative, often referred to as an indefinite integral, is essentially the reverse operation of differentiation. If you think about walking backwards, the antiderivative is sort of like that but for functions. It's about finding a function that, when differentiated, gives you the original function you started with.

For example, consider the function we're dealing with in the exercise, which is written as a simple equation, f(x) = 2. The antiderivative of this constant function is a line with a slope of 2, or graphically, it's like drawing a straight ramp that rises two units for every one unit it goes to the right. In mathematical terms, that's the function F(x) = 2x, plus some constant C. This constant can be any number because the slope of the line won't change regardless of where it starts on the Y-axis—it'll always have that consistent incline of 2.

Definite Integral

A definite integral is a fancy term for something pretty intuitive: it's a way to calculate the total accumulation of a quantity. Imagine you're filling up a swimming pool with water, and you measure how much water you've poured in at every moment. If you add up all those little bits of water, you get the total volume in the pool. A definite integral does something similar with functions—it adds up the values of the function at lots of tiny intervals over a certain range.

In our example, we use the definite integral to find the overall 'accumulation' under the line f(x) = 2 between x=1 and x=4. This process is made super easy thanks to the fundamental theorem of calculus, which tells us that we can simply evaluate the antiderivative we found at the upper limit F(4) and subtract the evaluation at the lower limit F(1) to get our answer: 6.

Area Under Curve

The phrase 'area under the curve' might sound technical, but it's just the space between the graph of our function and the x-axis, within a certain range. Think of it like the amount of paint you'd need to cover a floor if the edges were outlined by the curve of a function.

With this function f(x) = 2, the graph is a flat line, so the area under the curve over the interval from 1 to 4 is just a rectangle. In middle school math, you learned that the area of a rectangle is the width times the height, right? If we were to 'paint' our rectangle, we'd go three units wide—because that's our interval length from 1 to 4—and two units tall—the constant value of our function f(x).

Mathematical Proof

Mathematical proof is the process of showing that something is definitely true in the world of numbers and shapes. It's like a detective who collects evidence and presents it to a court to prove whodunnit—except, in this case, it's proving that the area under the curve is really what you say it is.

For the problem at hand, our 'detective work' involves two types of evidence: the definite integral and geometric calculation. First, we prove the area under the curve using the fundamental theorem of calculus, which is like saying we know the local ice cream store always sells one flavor daily because it's their unbreakable rule. Then we double-check our work by finding the area geometrically, which is similar to checking that the store indeed has that one flavor available today. Both methods lead us to the verdict: the area under the graph of f(x) = 2 from x=1 to x=4 is unequivocally 6 square units.