Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. $$ \int_{0}^{2}(x+4) d x $$

Step-by-Step Solution

Answer

The value of the definite integral \(\int_{0}^{2}(x+4) dx\) is 10. This represents the area under the curve \(y = x+4\) from \(x = 0\) to \(x = 2\).

Step 1: Find the Antiderivative

First, we find the antiderivative (F(x)) of the function \(x+4\). The antiderivative of \(x\) is \((1/2) x^2\) and the antiderivative of \(4\) is \(4x\). So, \(F(x) = (1/2) x^2 + 4x\).

Step 2: Evaluate the Antiderivative at the Limits of Integration

We then evaluate \(F(x)\) at \(x = 2\) and \(x = 0\) to find the definite integral. This gives us \(F(2) = (1/2) * 2^2 + 4*2 = 2 + 8 = 10\) and \(F(0) = (1/2) * 0^2 + 4*0 = 0\).

Step 3: Subtract the Two Results

According to the Fundamental Theorem of Calculus, the definite integral is obtained by subtracting the value of the antiderivative at the lower limit of integration from its value at the upper limit. This gives us \(10 - 0 = 10\).

Step 4: Graph the Area under the Curve

To graph the area under the curve, plot the original function \(x+4\) over the interval [0,2] and shade the area under the curve between x = 0 and x = 2. This area corresponds to the value of the definite integral.

Key Concepts

Antiderivative

The concept of an antiderivative is closely related to the process of finding the area under a curve.
Imagine taking the reverse approach of differentiation. That's what finding an antiderivative involves.
An antiderivative of a function is another function whose derivative is the original function.
For example, for the function \( f(x) = x+4 \),
its antiderivative is found by determining which function's derivative results in \( f(x) \).
In this case, the antiderivative, \( F(x) \), would be \((1/2)x^2 + 4x \).
Basically, the antiderivative gives us a broader picture — the family of all possible areas under the curve of \( f(x) \).

• To find antiderivatives, you integrate each term individually.
• The power rule (\( rac{1}{n+1} x^{n+1} \)) is commonly used for polynomial terms.
• Don't forget to add the constant of integration in indefinite integrals, though not needed in definite integrals.

Fundamental Theorem of Calculus

This theorem bridges the two central concepts in calculus: differentiation and integration.
Understanding it is crucial, as it provides a simple method to find definite integrals.
The theorem tells us that if we have a continuous function \( f(x) \) and its antiderivative \( F(x) \), then the definite integral over \( [a, b] \) is given by \( F(b) - F(a) \).

• Evaluate the antiderivative at the upper limit of integration: \( F(b) \).
• Evaluate the antiderivative at the lower limit of integration: \( F(a) \).
• Subtract: \( F(b) - F(a) \).

For the integral \( \int_{0}^{2}(x+4) \, dx \), using this theorem provides us with a precise computation of the area under \( f(x) \), from \( x=0 \) to \( x=2 \), giving us an answer of 10.

Graphing Utility

Graphing utilities are powerful tools that provide a visual aspect to mathematical concepts.
They can help you understand the areas represented by definite integrals better.
With graphing calculators or software, you can plot the function and observe the shaded area that corresponds to the integral. In our exercise, the function \( x+4 \) can be easily graphed.
Steps:

• Input the function \( x+4 \) into the graphing utility.
• Set the interval from \( x=0 \) to \( x=2 \).
• Shade the area under the curve in this interval.

This visual representation solidifies your understanding of integration as the accumulation of shade or area under a curve. Plus, these tools can also provide quick checks on your manual calculations, ensuring accuracy.