Problem 13 - Calculus Single Variable

Question

In Exercises \(11-20,\) calculate \(F(x)=\int_{a}^{x} f(t) d t\) $$ f(t)=4 t^{1 / 3} \quad a=8 $$

Step-by-Step Solution

Answer

\( F(x) = 3x^{4/3} - 48 \).

Step 1: Integrate the Function

To find the integral \( F(x) = \int_{a}^{x} f(t) \, dt \), we first need to integrate the function \( f(t) = 4t^{1/3} \). The integral of \( 4t^{1/3} \) with respect to \( t \) is found by using the power rule, which states that the integral of \( t^n \) is \( \frac{t^{n+1}}{n+1} \). Thus, \( \int 4t^{1/3} \, dt = 4 \cdot \frac{t^{1/3+1}}{1/3+1} = 4 \cdot \frac{t^{4/3}}{4/3} \).

Step 2: Simplify the Integral

Simplify the expression \( 4 \cdot \frac{t^{4/3}}{4/3} \). This can be rewritten as \( (4) \cdot \left( \frac{3}{4} \right) \cdot t^{4/3} = 3t^{4/3} \). So, the indefinite integral of \( 4t^{1/3} \) is \( 3t^{4/3} + C \), where \( C \) is the constant of integration.

Step 3: Evaluate the Definite Integral

To find \( F(x) \), we now need to evaluate the definite integral from \( a = 8 \) to \( x \), i.e., \( F(x) = \int_{8}^{x} 4t^{1/3} \, dt \). Substituting the definite bounds, we have \( F(x) = \left[ 3t^{4/3} \right]_{8}^{x} = 3x^{4/3} - 3(8)^{4/3} \).

Step 4: Calculate the Constant Term

Calculate \( 3(8)^{4/3} \). Since \( 8 = 2^3 \), \( 8^{4/3} = (2^3)^{4/3} = 2^4 = 16 \). Therefore, \( 3(8)^{4/3} = 3 \times 16 = 48 \).

Step 5: Write the Final Expression for F(x)

Now substitute back the constant term into the expression for \( F(x) \). Therefore, \( F(x) = 3x^{4/3} - 48 \).

Key Concepts

Power Rule in Integration

The power rule in integration is a fundamental technique used to find the integral of algebraic functions, specifically those in the form of \( t^n \). While differentiation has a power rule where you multiply by the exponent and decrease it by one, integration works as the reverse process.

To use the power rule for integration:

• Increase the exponent by one.
• Divide by the new exponent.

For example, to integrate \( 4t^{1/3} \), follow these straightforward steps:

• Increment the exponent: \( 1/3 + 1 = 4/3 \).
• Divide by the new exponent: \( \frac{t^{4/3}}{4/3} \). As you're also multiplying by a constant 4, it becomes \( 4 \cdot \frac{t^{4/3}}{4/3} \).
• Simplify to get \( 3t^{4/3} \), dismissing the constant of integration for indefinite integrals initially.

Understanding this rule makes integrating polynomial functions much easier and helps to prepare for more complex integration scenarios.

Integrating Functions

Integrating functions involves finding the antiderivative, or the reverse process of differentiation, of a given function. It is represented with the integral symbol \( \int \) and is the fundamental process in calculus to find areas under curves and solve many real-world problems.

The main steps for integrating functions look like this:

• Determine the function, here \( f(t) = 4t^{1/3} \).
• Apply the power rule for each term. For \( 4t^{1/3} \), use the power rule to conclude that the antiderivative is \( 3t^{4/3} \), from simplifying \( 4 \cdot \frac{t^{4/3}}{4/3} \).
• Handle constants and linear terms carefully by leaving them outside the integration.
  Finally, when integrating from a specific point \( a \) to another \( x \), this becomes a definite integral, giving a precise value, such as the area under the curve from 8 to \( x \) in this exercise.

Integrating functions is a key skill in solving areas, volumes, and more complex integrals that involve trigonometric or exponential functions.

Constant of Integration

The constant of integration, represented as "\( C \)," is an important part of indefinite integrals. When you integrate a function, the process of reversing differentiation naturally introduces a constant, since the derivative of a constant is zero.

Here's the breakdown:

• Every indefinite integral will include \( C \), because their antiderivatives can differ by a constant term without affecting the derivative.
• When you perform a definite integral, such as from \( 8 \) to \( x \) in this problem, the constant of integration effectively cancels out because you calculate using finite bounds, resulting in \( F(x) = 3x^{4/3} - 48 \).

The constant of integration is generally crucial in initial solutions, indicating that without specified bounds, there are infinite antiderivative functions, differing only by constant amounts. It also underscores how specifying initial conditions in calculus problems leads to fully determined solutions.