Problem 16 - Precalculus: Mathematical for Calculus

Question

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=4 x^{3}, \quad 2 \leq x \leq 5$$

Step-by-Step Solution

Answer

The area under the curve is 609.

Step 1: Understanding the Problem

We are asked to find the area under the curve of the function \(f(x) = 4x^3\) from \(x = 2\) to \(x = 5\). This involves calculating a definite integral over the specified interval.

Step 2: Set Up the Definite Integral

To find the area under the curve from \(x = 2\) to \(x = 5\), we set up the definite integral as follows: \[ \int_{2}^{5} 4x^3 \, dx. \]

Step 3: Find the Antiderivative

The next step is to find the antiderivative of \(4x^3\). An antiderivative \(F(x)\) of \(4x^3\) is \(F(x) = x^4\), since the derivative of \(x^4\) is \(4x^3\). Thus, \[ F(x) = x^4. \]

Step 4: Compute the Definite Integral

Now apply the Fundamental Theorem of Calculus, which involves evaluating the antiderivative at the upper and lower bounds and subtracting: \[ \int_{2}^{5} 4x^3 \, dx = [x^4]_2^5 = 5^4 - 2^4. \]

Step 5: Calculate the Values

Calculate the numerical values for \(5^4\) and \(2^4\): - \(5^4 = 625\) - \(2^4 = 16\)Thus, the definite integral results in: \(625 - 16 = 609.\)

Key Concepts

Antiderivative

The concept of an antiderivative is essential when solving problems involving definite integrals. Think of an antiderivative as the reverse of taking a derivative. While derivatives measure the rate at which things change, antiderivatives allow us to find the original function from its rate of change. In simpler terms, the antiderivative of a function is another function whose derivative gives the original function back.

For example, if the derivative of a function is given as \(4x^3\), an antiderivative would be \(x^4\). This is based on the power rule, where the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), provided \(n eq -1\). When integrating to solve the given exercise, this knowledge helps us transition from the rate of change back to the initial function's cumulative total.

Finding the antiderivative is the first step toward solving the definite integral, setting the foundation for applying the Fundamental Theorem of Calculus in the next stages.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with that of integration, acting as a bridge between these two key branches of calculus. It states that if you have a continuous function over an interval and an antiderivative of that function, then the definite integral of the function over that interval can be directly computed using the antiderivative.

The theorem simplifies the process of finding areas, such as those under curves, turning it into a matter of evaluating the antiderivative at the given limits of integration. For our problem, the definite integral of \(4x^3\) from \(x = 2\) to \(x = 5\) can be found by evaluating the antiderivative \(x^4\) at these points, as shown by the expression:- \([x^4]_2^5 = 5^4 - 2^4\)- This results in calculating the difference \(625 - 16\).

By applying this theorem, we convert the integral into an algebraic evaluation, efficiently determining the area under the curve.

Area Under a Curve

Finding the area under a curve is a common application of definite integrals in calculus. This area not only represents the sum of infinitely small rectangles under the curve but can also provide practical insights, such as the total distance traveled given a velocity-time graph.

For the function \(f(x) = 4x^3\) and the interval \(2 \leq x \leq 5\), the area we're interested in is precisely the definite integral:- \(\int_{2}^{5} 4x^3 \, dx\)

This integral calculates the sum of the values from \(x = 2\) to \(x = 5\), representing the enclosed area beneath the polynomial curve and the \(x\)-axis. Solving this tells us that the area is \(609\) square units. Understanding this concept is fundamental when dealing with real-world problems that can be graphically represented.

Polynomial Function

Polynomial functions, such as \(f(x) = 4x^3\), are expressions involving variables raised to various powers and are amongst the most basic yet versatile functions in calculus. These functions can be used to model a variety of real-world phenomena, from the motion of objects to economics and beyond.

In our problem, we are dealing specifically with a cubic polynomial, where the highest power of the variable \(x\) is 3. Such functions are continuous and differentiable, making them perfect candidates for integration. The integration of polynomial functions is straightforward, due to their predictable behavior and the power rule for integration, which allows us to easily find antiderivatives.

Understanding polynomial functions and their integration is crucial, as it forms the backbone of more complex problem-solving processes in calculus, helping students grasp the fundamental skills needed to tackle a wide range of mathematical challenges.