Problem 48 - Calculus: Early Transcendentals

Question

If \( \displaystyle \int^8_2 f(x) \, dx = 7.3 \) and \( \displaystyle \int^4_2 f(x) \, dx = 5.9 \), find \( \displaystyle \int^8_4 f(x) \, dx \).

Step-by-Step Solution

Answer

The integral \( \int_4^8 f(x) \, dx \) is 1.4.

Step 1: Understanding the Given Integrals

We are given two definite integrals: \( \int_2^8 f(x) \, dx = 7.3 \) and \( \int_2^4 f(x) \, dx = 5.9 \). These represent the areas under the curve \( f(x) \) from 2 to 8 and 2 to 4, respectively.

Step 2: Applying the Property of Definite Integrals

For any function \( f(x) \), the property \( \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx \) can be used to split or combine definite integrals. In this case, we use: \( \int_2^8 f(x) \, dx = \int_2^4 f(x) \, dx + \int_4^8 f(x) \, dx \).

Step 3: Solving for the Unknown Integral

Substitute the known values into the equation from the previous step: \( 7.3 = 5.9 + \int_4^8 f(x) \, dx \). Solve for \( \int_4^8 f(x) \, dx \) by subtracting 5.9 from both sides: \( \int_4^8 f(x) \, dx = 7.3 - 5.9 \).

Step 4: Calculating the Result

Subtract 5.9 from 7.3 to find \( \int_4^8 f(x) \, dx \). The calculation is: \( 7.3 - 5.9 = 1.4 \).

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on changes, motion, and the behavior of quantities. It consists of two main areas: differential calculus and integral calculus. While differential calculus looks at rates of change and slopes of curves, integral calculus focuses on finding the area under curves and the accumulation of quantities.

Calculus is essential in many fields including physics, engineering, and economics because it allows us to model dynamic systems and calculate changes over time. In our exercise, we are working with integral calculus to find definite integrals, which provide the total area under a chosen segment of the curve of a function. Understanding calculus concepts helps in interpreting data patterns and solving complex problems systematically.

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function. In simple terms, integration helps to calculate the total area under a curve described by a function. There are two primary types of integrals: indefinite and definite.

An indefinite integral, often represented as an antiderivative, does not have set limits and includes a constant of integration. On the other hand, a definite integral is calculated over a specific interval with upper and lower limits noted. It's represented as \( \int_a^b f(x) \, dx \), where \( a \) and \( b \) are the bounds.

In our scenario, given bounds help to target particular areas and calculate their size accurately. This process is crucial for extracting useful information about a function, such as total change or area under the curve within specified limits.

Properties of Definite Integrals

The properties of definite integrals simplify the evaluation of complex functions. One of the key properties is the additivity of the interval. This states that the integral from \( a \) to \( c \) can be split into two smaller integrals: from \( a \) to \( b \) and from \( b \) to \( c \). Mathematically, it is expressed as:
\[\int_a^c f(x) \, dx = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx\]

This property was used in solving the original exercise. We combined and separated the intervals to find the value of the unknown integral \( \int_4^8 f(x) \, dx \). Once we apply this property, calculating unknown integrals becomes easier using arithmetic operations involving known intervals.

Other notable properties include reversal of limits, which introduces a negative sign, and if two limits are the same, the integral equals zero. Understanding these characteristics allows one to manipulate integrals effectively for different problem sets.