Problem 35 - Applied Calculus

Question

Find an antiderivative. $$p(z)=(\sqrt{z})^{3}$$

Step-by-Step Solution

Answer

The antiderivative of \( p(z) = (\sqrt{z})^3 \) is \( \frac{2}{5} z^{5/2} + C \).

Step 1: Express the Function in Simplified Form

The given function is \( p(z) = (\sqrt{z})^3 \). We can rewrite \( \sqrt{z} \) as \( z^{1/2} \). Therefore, the expression becomes \( (z^{1/2})^3 = z^{3/2} \).

Step 2: Apply the Power Rule for Antiderivatives

Recall the power rule for finding antiderivatives: if \( f(x) = x^n \), an antiderivative is \( F(x) = \frac{x^{n+1}}{n+1} + C \), where \( C \) is an arbitrary constant of integration. For \( p(z) = z^{3/2} \), \( n = 3/2 \).

Step 3: Integrate Using the Power Rule

Apply the power rule to find the antiderivative of \( z^{3/2} \). It becomes \( \frac{z^{3/2 + 1}}{3/2 + 1} + C = \frac{z^{5/2}}{5/2} + C \).

Step 4: Simplify the Result

Simplify the expression \( \frac{z^{5/2}}{5/2} \). This is equivalent to \( \frac{2}{5} z^{5/2} \). So the antiderivative is \( \frac{2}{5} z^{5/2} + C \).

Key Concepts

Power Rule for Integration

When diving into integration, a crucial tool to have in your math toolkit is the Power Rule for Integration. This rule simplifies the process of finding antiderivatives, or the reverse operation of differentiation. Essentially, the Power Rule states:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]Here, \(C\) represents the constant of integration that accounts for any constant initially lost through differentiation. The rule is applicable provided \(n\) is not equal to -1, as this special case requires a different approach called the natural logarithm function. Using the Power Rule allows you to raise the exponent by one and divide by this new exponent. For instance, given a function \(x^3\), by applying the power rule, its integral equates to \(\frac{x^4}{4} + C\). This method makes integration straightforward and incredibly manageable. It is particularly useful in a wide array of mathematical problems involving polynomial expressions.

Exponents

Understanding exponents is vital in simplifying and solving integration problems. Exponents, also known as powers, are a way to denote repeated multiplication of a number by itself. For example, \(a^n\) implies that the base \(a\) is multiplied by itself \(n\) times. In the exercise we're reviewing, \(\sqrt{z}\) is represented as an exponent: \(z^{1/2}\). When the function \(p(z) = (\sqrt{z})^3\) is expressed as \(z^{3/2}\), it's achieved by multiplying the exponents: \((z^{1/2})^3 = z^{3/2}\). Grasp the rules of exponents as they help in transforming complex-looking problems into simpler, more standard forms. Some key rules include:

• Multiplying exponents: \((a^m)(a^n) = a^{m+n}\)
• Raising a power to a power: \((a^m)^n = a^{mn}\)
• Dividing exponents: \(a^m/a^n = a^{m-n}\)

These principles are powerful tools that can unravel complex algebraic expressions quickly.

Integration

Integration is the mathematical process of finding the antiderivative or the integral of a function, turning a function's rate of change back into the total change itself. It's essentially the reverse of differentiation. In practical terms, it allows us to determine the area under a curve, work backward from a rate of change to find a total quantity, or solve differential equations.In the given exercise, the objective is to find the antiderivative of \(p(z) = (\sqrt{z})^3\). By rewriting the function as \(z^{3/2}\) and using the Power Rule, we integrate to find:\[\int z^{3/2} \, dz = \frac{z^{5/2}}{5/2} + C = \frac{2}{5} z^{5/2} + C\]This result represents the accumulation of the function's rate of change \(z^{3/2}\) over its domain. The process of integration doesn't just stop at polynomial expressions. Instead, it's a fundamental concept in calculus with extensions into multiple integrals, trigonometric functions, and other areas of advanced mathematical analysis. Its practice is not only crucial for academic success but also for applications spanning science, engineering, and economics.