Antiderivative Of 1-(y/7)³ With Respect To Y A Step-by-Step Guide

In the realm of calculus, the antiderivative, also known as the indefinite integral, plays a pivotal role in solving a myriad of problems, ranging from finding areas under curves to modeling physical phenomena. This comprehensive guide delves into the process of finding the antiderivative of the expression 1 - (y/7)³ with respect to y, providing a step-by-step explanation along with illustrative examples to solidify your understanding.

Understanding Antiderivatives: The Foundation of Integral Calculus

Before embarking on the journey of finding the antiderivative of our specific expression, let's first lay a solid foundation by understanding the concept of antiderivatives. In essence, an antiderivative is the reverse operation of differentiation. If we have a function f(x), its antiderivative, denoted as F(x), is a function whose derivative is equal to f(x). Mathematically, this can be expressed as:

F'(x) = f(x)

The process of finding the antiderivative is called integration. The integral symbol ∫ is used to represent integration, and the expression ∫f(x) dx represents the indefinite integral of f(x) with respect to x. The 'dx' signifies that we are integrating with respect to the variable x.

Key Concepts to Grasp:

• The Power Rule: This rule states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1))/(n+1) + C, where C is the constant of integration.
• The Constant of Integration: When finding an antiderivative, we always add a constant of integration, denoted as 'C'. This is because the derivative of a constant is always zero, so there are infinitely many functions that could have the same derivative.
• Linearity of Integration: The integral of a sum or difference of functions is equal to the sum or difference of their individual integrals. Also, the integral of a constant times a function is equal to the constant times the integral of the function.

Step-by-Step Solution: Finding the Antiderivative of 1 - (y/7)³

Now that we have a firm grasp of the fundamental concepts, let's tackle the task of finding the antiderivative of 1 - (y/7)³ with respect to y. Here's a step-by-step breakdown:

Step 1: Expand the Expression

Our first step is to expand the expression (y/7)³:

(y/7)³ = y³/343

Therefore, our expression becomes:

1 - (y/7)³ = 1 - y³/343

Step 2: Apply the Linearity of Integration

Next, we apply the linearity of integration, which allows us to break down the integral into simpler parts:

∫(1 - y³/343) dy = ∫1 dy - ∫(y³/343) dy

Step 3: Integrate Each Term

Now, we integrate each term separately:

• ∫1 dy = y + C₁ (The antiderivative of a constant is the constant times the variable of integration)
• ∫(y³/343) dy = (1/343) ∫y³ dy (We can pull out the constant 1/343)

Applying the power rule to ∫y³ dy, we get:

∫y³ dy = (y⁴/4) + C₂

Therefore:

∫(y³/343) dy = (1/343) * (y⁴/4) + C₂ = y⁴/1372 + C₂

Step 4: Combine the Results and Add the Constant of Integration

Finally, we combine the results from Step 3 and add the constant of integration 'C':

∫(1 - y³/343) dy = y - y⁴/1372 + C

Therefore, the antiderivative of 1 - (y/7)³ with respect to y is y - y⁴/1372 + C, where C is the constant of integration.

Illustrative Examples: Solidifying Your Understanding

To further solidify your understanding, let's work through a couple of examples:

Example 1:

Find the antiderivative of 2x² + 3x - 1 with respect to x.

Solution:

1. Apply the linearity of integration:

∫(2x² + 3x - 1) dx = ∫2x² dx + ∫3x dx - ∫1 dx

2. Integrate each term using the power rule:

• ∫2x² dx = 2∫x² dx = 2(x³/3) + C₁ = (2x³/3) + C₁
• ∫3x dx = 3∫x dx = 3(x²/2) + C₂ = (3x²/2) + C₂
• ∫1 dx = x + C₃

3. Combine the results and add the constant of integration:

∫(2x² + 3x - 1) dx = (2x³/3) + (3x²/2) - x + C

Example 2:

Find the antiderivative of sin(x) - cos(x) with respect to x.

Solution:

1. Apply the linearity of integration:

∫(sin(x) - cos(x)) dx = ∫sin(x) dx - ∫cos(x) dx

2. Integrate each term (recall that the antiderivative of sin(x) is -cos(x) and the antiderivative of cos(x) is sin(x)):

• ∫sin(x) dx = -cos(x) + C₁
• ∫cos(x) dx = sin(x) + C₂

3. Combine the results and add the constant of integration:

∫(sin(x) - cos(x)) dx = -cos(x) - sin(x) + C

Common Mistakes to Avoid: A Word of Caution

When finding antiderivatives, it's crucial to be mindful of common mistakes that can lead to incorrect results. Here are a few pitfalls to avoid:

• Forgetting the Constant of Integration: Always remember to add the constant of integration 'C' when finding an indefinite integral. This is a common oversight that can significantly alter the solution.
• Misapplying the Power Rule: The power rule applies to x^n where n ≠ -1. Be careful not to apply it when n = -1, as the antiderivative of x⁻¹ is ln|x| + C.
• Incorrectly Applying the Linearity of Integration: Ensure that you correctly apply the linearity of integration by distributing the integral sign across all terms in the expression.

Conclusion: Mastering the Art of Antiderivatives

Finding antiderivatives is a fundamental skill in calculus with wide-ranging applications. By understanding the basic concepts, applying the rules of integration, and avoiding common mistakes, you can master the art of finding antiderivatives and unlock a world of mathematical possibilities. This guide has provided a comprehensive explanation of how to find the antiderivative of 1 - (y/7)³ with respect to y, along with illustrative examples and cautionary notes. Remember to practice regularly to hone your skills and deepen your understanding. With dedication and perseverance, you'll become proficient in the realm of antiderivatives and integral calculus. Mastering antiderivatives opens doors to various fields, from physics and engineering to economics and statistics. The ability to find antiderivatives allows us to model real-world phenomena, solve differential equations, and analyze data with greater precision. So, embrace the challenge, delve into the intricacies of integration, and unlock the power of antiderivatives!