Problem 32 - Calculus Single Variable

Question

Evaluate the given definite integral by finding an antiderivative of the integrand and applying Theorem \(3 .\) $$ \int_{1}^{16} x^{-3 / 4} d x $$

Step-by-Step Solution

Answer

The integral evaluates to 4.

Step 1: Identify the integral

The given integral is \( \int_{1}^{16} x^{-3/4} \, dx \). We need to find the antiderivative (or indefinite integral) of the integrand \( x^{-3/4} \).

Step 2: Find the antiderivative

To find the antiderivative, we use the power rule for integration: if \( \int x^n \ dx = \frac{x^{n+1}}{n+1} + C \) (for \( n eq -1 \)), then for \( x^{-3/4} \), the antiderivative is \( \int x^{-3/4} \, dx = \frac{x^{-3/4+1}}{-3/4+1} = \frac{x^{1/4}}{1/4} + C = 4x^{1/4} + C \).

Step 3: Apply the Fundamental Theorem of Calculus Part 2

The Fundamental Theorem of Calculus Part 2 tells us that if \( F \) is an antiderivative of \( f \), \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \). Here, \( F(x) = 4x^{1/4} \), so we need to evaluate \( F(16) - F(1) \).

Step 4: Evaluate the antiderivative at the bounds

Calculate \( F(16) \) and \( F(1) \):- \( F(16) = 4(16)^{1/4} = 4 \times 2 = 8 \)- \( F(1) = 4(1)^{1/4} = 4 \times 1 = 4 \)

Step 5: Subtract to find the result

Subtract \( F(1) \) from \( F(16) \):- Result = \( F(16) - F(1) = 8 - 4 = 4 \).

Key Concepts

Antiderivative

The antiderivative, also known as the indefinite integral, is central in calculus. It's essentially the reverse of taking a derivative. When you find the antiderivative, you are determining which function, when differentiated, will yield the original function given in the integral.
\( \int x^{-3/4} \, dx \) seeks to find a function whose derivative is \( x^{-3/4} \).
The antiderivative includes a constant \( C \) because derivatives of constant terms are zero, making them invisible in differentiation.

1. Think of the antiderivative as finding the 'parent' function.
2. This process is a fundamental skill in solving indefinite integrals.

Understanding this concept well will make it easier to solve definite integrals through the Fundamental Theorem of Calculus.

Power Rule for Integration

The Power Rule for Integration is a simple yet powerful tool for finding antiderivatives, applicable when dealing with polynomial expressions.
The rule is defined as: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), for any real number \( n eq -1 \).
In the exercise, the integrand is \( x^{-3/4} \).

• First, increase the exponent \( -3/4 \) by one, which gives \( 1/4 \).
• Then, divide by this new exponent, resulting in \( \frac{x^{1/4}}{1/4} \).
• Finally, rewrite this as \( 4x^{1/4} + C \).

This rule allows us to seamlessly find the antiderivative necessary to apply in further integration processes.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of the derivative and the integral, showing that they are inverse processes.
It consists of two main parts, but here we focus on Part 2, which links definite integrals to antiderivatives.
It states: If \( F \) is an antiderivative of \( f \), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).

• This means to evaluate a definite integral, compute the antiderivative \( F(x) \), and use it to find \( F(b) - F(a) \).
• For \( \int_{1}^{16} x^{-3/4} \, dx \), the antiderivative \( F(x) \) is \( 4x^{1/4} \).
• Evaluate at the limits: \( F(16) = 8 \) and \( F(1) = 4 \).
• Subtract these to get the integral's value: \( 8 - 4 = 4 \).

Hence, the theorem simplifies calculating definite integrals by using antiderivatives.