Problem 100 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{1}^{2} x^{5 / 2} d x $$

Step-by-Step Solution

Answer

The definite integral evaluates to \( \frac{8\sqrt{2} - 2}{7} \).

Step 1: Set Up the Definite Integral

We need to evaluate the definite integral \[ \int_{1}^{2} x^{5/2} \, dx \]This involves finding the antiderivative of the integrand and then using the limits of integration.

Step 2: Find the Antiderivative

The function inside the integral is \( x^{5/2} \). To find its antiderivative, use the rule for integrating power functions: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]For our function, \( n = \frac{5}{2} \), so the antiderivative is: \[ \frac{x^{7/2}}{7/2} = \frac{2}{7} x^{7/2} \]

Step 3: Evaluate the Antiderivative at the Upper Limit

According to the Fundamental Theorem of Calculus, we need to evaluate the antiderivative \( \frac{2}{7} x^{7/2} \) at the upper limit, which is 2:\[ \frac{2}{7}(2^{7/2}) \] Calculate:\[ 2^{7/2} = (2^3)^{1/2} imes 2^{1/2} = 4\sqrt{2} \]Thus,\[ \frac{2}{7} \times 4\sqrt{2} = \frac{8\sqrt{2}}{7} \]

Step 4: Evaluate the Antiderivative at the Lower Limit

Next, evaluate the antiderivative at the lower limit, which is 1:\[ \frac{2}{7}(1^{7/2}) = \frac{2}{7} \]

Step 5: Subtract Lower Limit Evaluation from Upper Limit Evaluation

Finally, subtract the evaluation at the lower limit from the evaluation at the upper limit according to:\[ F(2) - F(1) = \frac{8\sqrt{2}}{7} - \frac{2}{7} \]Combine the terms:\[ = \frac{8\sqrt{2} - 2}{7} \]

Key Concepts

Antiderivative

An antiderivative of a function is akin to its reverse derivative. It essentially takes us back to the original function from its derivative form. When dealing with integrals, finding the antiderivative is a crucial step.
This is often referred to as "indefinite integration."

In the context of the given problem, the function given inside the integral is the power function, \( x^{5/2} \). To determine its antiderivative, we use the power rule for integration.

• General Rule: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
  
• Here, \( n = 5/2 \), so the antiderivative will be:
  \[ \frac{x^{7/2}}{7/2} = \frac{2}{7}x^{7/2} \]
  

The constant \( C \) doesn't play a direct role in definite integrals, as the evaluation cancels it out, but it's essential in indefinite integrals. Understanding this step is vital; it sets the stage for solving the definite integral through the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is the bridge between differentiation and integration. It establishes a connection that allows us to evaluate the definite integrals easily through antiderivatives.
The theorem essentially states two things:

• The derivative of an integral (of a function over an interval) gives back the original function.
  
• The integral of a function can be found by evaluating its antiderivative at the endpoints of the interval and subtracting these values.
  

In simpler terms, to compute a definite integral \( \int_{a}^{b} f(x) \, dx \) using the antiderivative \( F(x) \) of the function \( f(x) \), you use:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
For our exercise, the antiderivative \( \frac{2}{7}x^{7/2} \) is evaluated at the upper limit 2 and the lower limit 1 and then subtracted:

• Upper limit evaluation: \( \frac{2}{7}(2^{7/2}) \)
• Lower limit evaluation: \( \frac{2}{7}(1^{7/2}) \)
• Result: \( \frac{8\sqrt{2} - 2}{7} \)
  

Mastery of this theorem is crucial for evaluating definite integrals efficiently.

Power Rule for Integration

The power rule for integration is a fundamental tool for solving integrals involving polynomial functions.
It is particularly useful when dealing with expressions where the variable is raised to a power. The power rule enables us to find the antiderivative rather effortlessly.

The power rule states:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( n \) is any real number except \( -1 \). When applying this rule:

• Increase the exponent by one: from \( n \) to \( n+1 \).
  
• Divide by the new exponent: \( n+1 \).
  
• Add a constant of integration \( C \) for indefinite integrals.
  

In our specific case, \( n = 5/2 \). By following the rule, the antiderivative of \( x^{5/2} \) becomes:
\[ \frac{2}{7}x^{7/2} \]
This method simplifies what would otherwise require more complex or numerous calculations, making it invaluable for calculus students.