Problem 17 - Calculus Early Transcendentals

Question

Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$f(y)=-2 / y^{3}$$

Step-by-Step Solution

Answer

Answer: The antiderivative of the function $$f(y) = -2/y^{3}$$ is $$F(y) = \frac{1}{y^{2}} + C$$.

Step 1: Recognize the given function

Recognize the given function $$f(y)=-2 / y^{3}$$ which we need to integrate.

Step 2: Rewrite the function for integration

Rewrite the given function with respect to y: $$\int -2 y^{-3} dy$$

Step 3: Integrate the function

Apply the integration rule which states that $$\int x^{n}dx = \frac{x^{n+1}}{n+1}+C$$, where n is a constant and C is the constant of integration. So, we have: $$\int -2 y^{-3} dy = -2 \int y^{-3} dy = -2 \cdot \frac{y^{-3+1}}{-3+1}+C$$

Step 4: Simplify the result

Now, simplify the expression: $$-2 \cdot \frac{y^{-2}}{-2}+C = y^{-2} + C$$

Step 5: Rewrite the result in a more readable form

Rewrite the antiderivative by replacing $$-2$$ in the exponent with a division notation: $$y^{-2}+C=\frac{1}{y^{2}} + C$$

Step 6: Check the result by taking the derivative

Take the derivative of the antiderivative to verify if it equals the original function: $$\frac{d}{dy}(\frac{1}{y^{2}} + C) = -2\cdot y^{-3} = -2/y^{3}$$ The derivative of the antiderivative matches the original function, so the antiderivative of $$f(y) = -2 / y^{3}$$ is $$F(y) = \frac{1}{y^{2}} + C$$.

Key Concepts

Integration

Integration is a fundamental concept in calculus that represents the process of finding the antiderivative or indefinite integral of a function. It is the reverse operation of differentiation, where instead of finding the rate of change (derivative), we are looking for the original function that was differentiated. Think of it as a puzzle where derivatives give you pieces, and integration is about putting those pieces back together to see the whole picture.

When integrating a function, such as f(y) = -2 / y^3, we look for a function F(y) such that F'(y) = f(y). This reconstructed function F(y) is then called the antiderivative of f(y). Integration is crucial in mathematics as it allows for the determination of areas, volumes, and other concepts integral to fields like physics, engineering, and economics.

Power Rule for Integration

The power rule for integration is a handy tool for finding antiderivatives of functions that can be expressed as x^n, where n is any real number except -1. The rule states that the integral of x^n with respect to x is (x^{n+1}) / (n+1), and then don't forget the '+ C', where C is the integration constant.

For instance, if we apply the power rule to the function f(y) in our example, which is -2 * y^{-3}, we would increase the exponent by 1 to get (-3 + 1), and divide by this new exponent, giving us the antiderivative (1 / y^2) + C. This rule simplifies the integration process, making it one of the essential techniques in calculus.

Integration Constants

Integration constants are an integral part of finding the antiderivatives. Whenever we integrate a function, we add an arbitrary constant, denoted by C. This constant reflects the fact that there are an infinite number of possible antiderivatives for a function, as the derivative of a constant is zero.

In the exercise given, after applying the power rule for integration to find the antiderivative of -2 / y^3, we add the constant C to denote that any function of the form (1 / y^2) + C could be the antiderivative of f(y). This accounts for all the parallel curves that differ by a vertical shift, emphasizing the importance of considering initial conditions or additional information to solve for C when dealing with definite integrals or initial value problems.