Problem 53 - Calculus

Question

Find \(f(x)\) if \(\int_{0}^{x} f(t) d t=\frac{1}{3} x^{3}\)

Step-by-Step Solution

Answer

The function is \( f(x) = x^2 \).

Step 1: Understand the Relationship

The problem states that the integral \( \int_{0}^{x} f(t) \, dt = \frac{1}{3} x^3 \). This integral represents a function in terms of \( x \), therefore, we can differentiate both sides with respect to \( x \) to find \( f(x) \).

Step 2: Differentiate Both Sides

To find \( f(x) \), differentiate both sides of the equation with respect to \( x \). The left side becomes \( f(x) \) due to the Fundamental Theorem of Calculus. The right side, after differentiation, is \( x^2 \).

Step 3: Write the Expression for \( f(x) \)

After differentiating, we find that \( f(x) = x^2 \). This is because the derivative of \( \frac{1}{3}x^3 \) with respect to \( x \) is \( x^2 \).

Key Concepts

Integration

Integration is a core concept in calculus that involves finding the area under a curve or the accumulation of quantities. Understanding this process is crucial as it provides insights into how one can reverse the process of differentiation. Think of integration as a way to "sum up" many tiny rectangles under a curve to find the total area. In more formal terms, if you have a function \( f(t) \), the integral \( \int f(t) \, dt \) gives you the "accumulated quantity" associated with that function.

There are different types of integrals, but in this context, we're dealing with the definite integral. When we take the definite integral of \( f(t) \) from 0 to \( x \), denoted as \( \int_{0}^{x} f(t) \, dt \), we're finding the net area between the function \( f(t) \) and the \( t \)-axis from \( t=0 \) to \( t=x \).

• The result of a definite integral is a number that represents the total accumulation over that interval.
• This process plays a key role in various real-world applications like calculating distances, areas, and even probabilities in statistics.

Differentiation

Differentiation is another fundamental calculus concept, often seen as the counterpart to integration. When you differentiate a function, you're essentially finding how quickly or slowly that function is changing. It's about discovering the rate at which one quantity changes with respect to another. In more simple terms, differentiation helps us find the slope of the tangent line to the curve of a function at any given point.

In the context of the problem, when we take the derivative of the given integral \( \int_{0}^{x} f(t) \, dt = \frac{1}{3} x^3 \), we apply differentiation to both sides of the equation. This step helps to "undo" the integration and directly find the function \( f(x) \).

• Differentiating the left side yields \( f(x) \) based on the Fundamental Theorem of Calculus.
• The derivative of the right side, \( \frac{1}{3} x^3 \), is calculated using basic differentiation rules, resulting in \( x^2 \).

Definite Integral

A definite integral is an integral with specified upper and lower limits, which in our problem are given as 0 and \( x \). This tells us the exact "start" and "end" points for evaluating the area under a curve. More formally, if \( F(x) \) is the antiderivative of \( f(t) \), then the definite integral from \( a \) to \( b \) is \( F(b) - F(a) \).

In this exercise, when we see \( \int_{0}^{x} f(t) \, dt \), we understand that it's giving us a function specifically evaluated from point 0 to \( x \).

• Finding a definite integral allows us to calculate the net change or total accumulation, such as the total distance travelled given a velocity function.
• By solving for \( f(x) \) through differentiation, we utilize the fact that the definite integral encapsulates information about the accumulation up to that point in \( x \).

The use of the definite integral in the problem is crucial as it gives a concrete value which can then be manipulated through differentiation to quickly find the original function.