Problem 23 - Applied Calculus

Question

Use the Fundamental Theorem to determine the value -of \(b\) if the area under the graph of \(f(x)=x^{2}\) between \(x=0\) and \(x=b\) is equal to \(100 .\) Assume \(b > 0\)

Step-by-Step Solution

Answer

\( b = \sqrt[3]{300} \) (approximately 6.7)

Step 1: Identify the given problem

We need to find a value of \( b \) such that the area under the curve of the function \( f(x) = x^2 \) from \( x = 0 \) to \( x = b \) equals 100.

Step 2: Express the area as a definite integral

The area under the curve \( f(x) = x^2 \) from \( x = 0 \) to \( x = b \) can be expressed as the definite integral \( \int_{0}^{b} x^2 \, dx \).

Step 3: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \( F(x) \) is an antiderivative of \( f(x) \), then\[ \int_{0}^{b} f(x) \, dx = F(b) - F(0). \] So, we need to find an antiderivative of \( f(x) = x^2 \).

Step 4: Find the antiderivative of \( x^2 \)

The antiderivative of \( x^2 \) is \( \frac{x^3}{3} + C \). For a definite integral, the constant \( C \) is not needed, so we have \( F(x) = \frac{x^3}{3} \).

Step 5: Evaluate the definite integral

Using the antiderivative, we evaluate:\[ \int_{0}^{b} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{b} = \frac{b^3}{3} - \frac{0^3}{3} = \frac{b^3}{3}. \]

Step 6: Set the integral equal to 100

Since the given area is 100, we set the integral equal to 100:\[ \frac{b^3}{3} = 100. \]

Step 7: Solve for \( b \)

Multiply both sides by 3 to clear the fraction:\[ b^3 = 300. \] Take the cube root of both sides to solve for \( b \):\[ b = \sqrt[3]{300}. \]

Key Concepts

Definite Integral

The concept of a definite integral is all about finding the area under a curve between two points on the x-axis. This integral takes a function and calculates the accumulation of quantities, such as areas, between specific limits.

A definite integral is written in the form \( \int_{a}^{b} f(x) \, dx \), where:

• \( f(x) \) is the function we are integrating, or the curve under which we want to calculate the area
• \( a \) and \( b \) are the limits of integration, representing the interval on the x-axis
• \( dx \) indicates the variable of integration

This form tells us to integrate the function \( f(x) \) from \( x = a \) to \( x = b \). The result is a single numerical value, which represents the total area under the curve from \( x = a \) to \( x = b \). These integrals are incredibly useful for solving problems related to total areas or accumulated values in calculus.

Antiderivative

An antiderivative is a function whose derivative is the original function we started with. When we take the antiderivative of a function, we essentially reverse the process of differentiation.

Let's say we have a function \( f(x) = x^2 \). The antiderivative of this function is a new function \( F(x) \), such that\( \frac{d}{dx}[F(x)] = f(x) \).

In our original solution, the antiderivative of \( x^2 \) found was \( \frac{x^3}{3} + C \), where \( C \) is a constant of integration. In definite integrals, however, this constant cancels out, making calculations simpler as \( C \) is not needed for the final answer.

Antiderivatives are crucial when applying the Fundamental Theorem of Calculus, which links differentiation and integration in one elegant framework.

Area Under a Curve

Finding the area under a curve of a function is a common objective in calculus, leading to crucial applications in various fields, such as physics and engineering.

The area is calculated between two points on the x-axis for the particular function you're examining. This area is numerically equivalent to a definite integral. In many applications, finding this area helps determine quantities like distance, probability, or in this instance, the needed value of \( b \) in the original exercise.

For specific problems, like calculating the area under \( f(x) = x^2 \), the definite integral from \( x = 0 \) to \( x = b \) gives \( \int_{0}^{b} x^2 \, dx \). Solving this integral gives us a specific numeric value that represents the area sought.