Question

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{0} 3 d x $$

Short answer

The area represented by the integral is 3 units squared.

Step-by-step solution

Identify the function

The integral \(\int_{0}^{3} dx\) is signifying the area under the curve from 0 to 3 of the function \(f(x) = 1\). This is because the 'dx' term is alone in the integral, meaning the function we’re integrating is simply 1.

Sketch the region

To sketch this, draw a rectangle with base from 0 to 3 on x-axis and a constant height of 1 (the value of our function). This rectangle represents the area under the curve of the function \(f(x) = 1\) from 0 to 3.

Evaluate the integral

The geometric area of a rectangle is given by the formula: base x height. In this case, the base is 3 (upper limit - lower limit) and the height is 1 (the value of our constant function). So, the area under the curve (or the integral) is \(3*1 = 3\).

Key concepts

Integration Techniques

When facing the task of evaluating a definite integral, it's crucial to apply effective integration techniques. Definite integrals represent the accumulation of a quantity, and finding their value often requires strategic approaches.

Some common integration techniques include the power rule, substitution (also known as u-substitution), integration by parts, partial fraction decomposition, and trigonometric integration. Each of these methods can be selected based on the form of the function being integrated. For the integral \( \int_{0}^{3} dx \), we see that the function is a constant, \(f(x) = 1\), and does not require complex techniques. In such simple cases, knowledge of basic geometric shapes and their areas can suffice to evaluate the integral directly.

Complex functions might not offer such straightforward solutions and require the extended toolkit of integration methods. By mastering these techniques, one ensures a method is available for different types of functions, whether they are polynomials, exponentials, or trigonometric functions.

Geometric Interpretation of Integrals

The geometric interpretation of a definite integral is one of the most visually powerful concepts in calculus. It allows us to visualize the problem and find areas even without performing algebraic manipulations. The definite integral of a function between two points on the x-axis is equivalent to the area under the curve of that function between those points.

For the integral \( \int_{0}^{3} dx \), interpreting it geometrically means identifying that we're looking for the area under the curve \(f(x) = 1\) from \(x = 0\) to \(x = 3\). Since \(f(x) = 1\) represents a constant function, we're effectively finding the area of a rectangle. This geometric relationship transforms an abstract calculus problem into a simple geometry problem, making it far more approachable.

The beauty of this interpretation is that it ties together algebraic procedures with spatial understanding, reinforcing the connection between mathematics and the physical world. For example, integrals can represent accumulated distances, area between curves, volumes, and more, illustrating the versatility of geometric interpretation.

Area Under a Curve

The concept of 'area under a curve' captures the essence of definite integrals in a physical sense. When you calculate the definite integral of a function, what you're really doing is measuring the total 'size'—in terms of area—of the region that lies between the graph of the function and the x-axis, within certain boundaries.

For the given exercise, the area under the curve is simple to find. The function \(f(x) = 1\) is a horizontal line, and the area beneath it from \(x = 0\) to \(x = 3\) is a rectangle. The integral \( \int_{0}^{3} dx \) is evaluated by multiplying the width of the interval (which is 3 units, from the limits of integration) with the height of the function (constant at 1 unit). This yields an area of 3 square units.

Understanding this concept is fundamentally important as it extends beyond flat shapes to more complex functions, including those that curve and twist above and below the x-axis. Beyond mathematics, 'area under a curve' has real-world applications in physics, economics, and other sciences, where it can represent things like displacement, profits, or likelihoods, demonstrating the concept's diverse implications.