Problem 51 - Applied Calculus

Question

Evaluate each definite integral. $$ \int_{1}^{4} \frac{1}{y^{2}} d y $$

Step-by-Step Solution

Answer

\( \frac{3}{4} \)

Step 1: Set Up the Integral

To evaluate the definite integral \( \int_{1}^{4} \frac{1}{y^{2}} \, d y \), first recognize that \( \frac{1}{y^2} \) can be rewritten as \( y^{-2} \). This is done to make it easier to integrate.

Step 2: Find the Antiderivative

Since \( \int y^n \, dy = \frac{y^{n+1}}{n+1} + C \), we apply this to \( \int y^{-2} \, dy \). Here, \( n = -2 \), therefore, the antiderivative is \( \int y^{-2} \, dy = \frac{y^{-1}}{-1} + C = -\frac{1}{y} + C \).

Step 3: Evaluate the Antiderivative at the Upper Limit

Plug the upper limit of the integral, 4, into the antiderivative: \( -\frac{1}{4} \).

Step 4: Evaluate the Antiderivative at the Lower Limit

Plug the lower limit of the integral, 1, into the antiderivative: \( -\frac{1}{1} = -1 \).

Step 5: Calculate the Definite Integral

Subtract the result of the antiderivative at the lower limit from the result at the upper limit: \( -\frac{1}{4} - (-1) = -\frac{1}{4} + 1 = \frac{3}{4} \).

Key Concepts

Understanding the Antiderivative

The concept of antiderivatives is essential when evaluating integrals. An antiderivative of a function is, in simple terms, a function whose derivative returns the original function. So, when you integrate, you are essentially finding this function.
In our exercise, we need to find the antiderivative of \( \frac{1}{y^2} \) or \( y^{-2} \). Remember, the basic formula to find an antiderivative, or indefinite integral, is \( \int y^n \, dy = \frac{y^{n+1}}{n+1} + C \), where \( C \) is the constant of integration because integration is the inverse of differentiation. However, when dealing with definite integrals, this constant \( C \) cancels out later in the process.
Once we identify \( n = -2 \), we apply the formula: \( \int y^{-2} \, dy = \frac{y^{-1}}{-1} + C = -\frac{1}{y} + C \). This provides us with an expression to use when we evaluate the integral with the given limits.

Exploring Limits of Integration

Limits of integration are the values that define the interval over which you're calculating the area under a curve in a definite integral. For indefinite integrals, this area stretches to infinity, but definite integrals limit this area to a specific range.
In the problem provided, the limits are 1 and 4. These values are crucial as they determine where you start and stop when calculating the integral. Plug these limits into the antiderivative equation to find the definite integral.
The process involves evaluating the antiderivative \( -\frac{1}{y} \) at these two points. First, substitute the upper limit (4) into the antiderivative, yielding \( -\frac{1}{4} \). Next, substitute the lower limit (1) to get \( -1 \). With these calculations, we can proceed to find the integral's value.

Integrating Powers of y Simplified

Integrating powers of \( y \) can initially seem challenging, but simplified rules make it easier. Each time you need to integrate a power like \( y^n \), remember the power rule for integrals: \( \int y^n \, dy = \frac{y^{n+1}}{n+1} \), applicable when \( n eq -1 \).
By rewriting \( \frac{1}{y^2} \) as \( y^{-2} \), we transform it into a format compatible with our easy-to-use power rule. This rule is the key that simplifies the process of integration.
Once you have the antiderivative result for basic powers, apply the limits of integration to find your definite result. The transformation of \( \frac{1}{y^2} \) to \( y^{-2} \) simplifies finding \( \int_{1}^{4} \frac{1}{y^2} \, dy \), resulting in the solution of \( \frac{3}{4} \) effortlessly.