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Magazine Chapter 5: Problem 29
\(21-44\) Evaluate the integral. $$\int_{-2}^{-1}\left(4 y^{3}+\frac{2}{y^{3}}\right) d y$$
Short Answer
Expert verified
The evaluated integral is -15.75.
Step by step solution
01
Identify the components of the integral
The integral given is \( \int_{-2}^{-1} \left(4y^3 + \frac{2}{y^3}\right) \, dy \). It is composed of two separate functions: \( 4y^3 \) and \( \frac{2}{y^3} \). We will integrate each term separately.
02
Integrate the first term
The integral of \( 4y^3 \) with respect to \( y \) is: \[ \int 4y^3 \, dy = \frac{4}{4}y^4 = y^4 \] evaluated at \([-2, -1]\).
03
Integrate the second term
For the second component \( \frac{2}{y^3} \), rewrite it as \( 2y^{-3} \). The integral is: \[ \int 2y^{-3} \, dy = \frac{2}{-2}y^{-2} = -y^{-2}\] evaluated at \([-2, -1]\).
04
Evaluate the integrated expressions from -2 to -1
Evaluate \( y^4 \) from -2 to -1: \[ [-1]^4 - [-2]^4 = 1 - 16 = -15 \] Evaluate \( -y^{-2} \) from -2 to -1: \[ -[-1]^{-2} - (-[-2]^{-2}) = -1 + \frac{1}{4} = -\frac{3}{4} \].
05
Sum the evaluated results
To find the value of the definite integral, add the results of both components:\(-15 + (-\frac{3}{4}) = -15 - \frac{3}{4} = -15.75\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral Calculus
Integral calculus is a fundamental part of mathematics, crucial for many applications in science and engineering. It focuses on the concept of integration, an operation that calculates the accumulated total or area under a curve. This process is essentially the reverse of differentiation, which deals with rates of change.
Key principles of integral calculus include:
When you work with integrals, you're often concerned with finding an antiderivative, which is a function whose derivative matches the function you're integrating.
Key principles of integral calculus include:
- Finding areas, volumes, and other quantities accumulated over a range.
- Analyzing functions to determine the integral or total change over a particular interval.
When you work with integrals, you're often concerned with finding an antiderivative, which is a function whose derivative matches the function you're integrating.
Evaluation of Integrals
Evaluating integrals, as demonstrated in the exercise, involves calculating the area under a curve for a specific interval. The definite integral \[ \int_{a}^{b} f(x) \, dx \]measures this area from point \(a\) to point \(b\). In our example, \[\int_{-2}^{-1} (4y^3 + \frac{2}{y^3})\, dy\]we evaluate each part separately.
Steps in evaluating integrals often include:
Steps in evaluating integrals often include:
- Identifying components of the integrand that can be easily integrated.
- Using known antiderivatives or integration rules.
- Evaluating the antiderivative at the boundaries of the integral.
- Subtracting the values obtained at each boundary to find the total area or accumulated change.
Antiderivatives
Antiderivatives, also known as indefinite integrals, are pivotal in solving integrals. The antiderivative of a function is a new function whose derivative gives back the original function. This is like "undoing" the differentiation process.
To find the antiderivative, you might use these common rules:
To find the antiderivative, you might use these common rules:
- For a constant multiplied by a function, like \( c f(x) \), integrate each part separately: \( \int c \, f(x) \, dx = c \int f(x) \, dx \).
- A polynomial like \( y^n \) can be integrated by: \( \int y^n \, dy = \frac{y^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
Step by Step Integration
Step by step integration is a detailed process ensuring you correctly follow the rules of integration to arrive at the right solution. This involves a systematic method to tackle the definite integral:
- Identification: Recognize different parts or terms in your function to treat them individually.
- Integration: Apply integration rules to find the antiderivative of each term independently.
- Evaluation: Substitute the limits of integration into the resulting expression.
- Synthesis: Add or subtract the values found to compute the final result.
