Calculating the Area between Limits with Definite Integrals

To determine the area enclosed between two points on a continuous function within a closed interval [a, b], we can use a definite integral. This type of integral will result in a specific numerical value, unlike an indefinite integral which produces a function.

Notation for Definite Integrals

Definite integrals are written similarly to indefinite integrals, but with the addition of the limits appearing as subscript and superscript on the integration sign. For example, if we want to integrate a function between the limits of 5 and 8, the notation would be:

• ∫⁸₅ f(x) dx

Solving Definite Integrals Using a Specific Method

When solving definite integrals, there is a specific procedure that must be followed:

• 1) Write the integral: Begin by writing the definite integral with its limits in the form ∫^b_a f(x) dx.
• 2) Integrate the function: Follow the same steps as an indefinite integral to find the function f(x). Do not include the integration constant, C, in the result. Rewrite the answer as [f(x)]^b_a.
• 3) Evaluate f(x) between the given limits: By plugging in the limits, we can find the final value of the definite integral by subtracting f(b) from f(a).

Some may wonder why the integration constant is not included in the evaluation of f(x). Consider if we did include C, and called it g(x). Then, the value of g(x) = f(x) + C. When finding the value of g(x) between the given limits, we would have g(b) - g(a) = (f(b) + C) - (f(a) + C) = f(b) - f(a). As we can see, the integration constant cancels itself out, which is why we do not include it in the calculations.

Finding the Area Using Definite Integrals

Definite integrals are a useful tool for finding the area under a curve. For example, if we want to determine the area enclosed between the x-axis and the curve f(x) = 5x² between x = 1 and x = 7, we can represent this situation graphically.

Graph showing f(x) = 5x² with limits x = 1 and x = 7, Nilabhro Datta - StudySmarter Originals

• ∫⁷₁ 5x² dx
• = (5/3 × 7³) - (5/3 × 1³)
• = 570

Finding the Area Under the Curve

In the previous example, we found the area enclosed between the curve and the x-axis by calculating the definite integral. We can visualize this on a graph showing the area filled by the curve.

Graph showing f(x) = 5x² with limits x = 1 and x = 7, Nilabhro Datta - StudySmarter Originals

Evaluating Definite Integrals Using Radians

Sometimes, the units of measurement for the given limits may not be in terms of x. In these cases, we must first convert the units before evaluating the definite integral. For instance, let's consider the curve y = cos(x) and find the area enclosed between x = 0.5 and x = 1.

Note: x is in radians

• ∫¹_0.5 cos(x) dx
• = sin(1) - sin(0.5)
• = 0.841-0.479
• = 0.362

Similarly to the previous example, the value for the definite integral may be negative if the enclosed area falls below the x-axis. This is depicted on the following graph.

Graph showing f(x) = cos(x) with limits x = 0.5 and x = 1, Nilabhro Datta - StudySmarter Originals

Interpreting Negative Areas

If the area enclosed by the curve and the x-axis falls below the x-axis, the value for the definite integral will be negative. On the other hand, a positive value indicates that the area is above the x-axis. If we need to find the total magnitude of the enclosed area, including both positive and negative areas, we must find the individual areas and add their magnitudes without considering their signs.

Finding the Total Area Enclosed by a Curve

To find the area enclosed by the curve y = x (x - 5) and the x-axis, we first need to determine the points where the curve intersects the axis.

To find the area bounded by the curve and the x-axis:

• 1) Determine the points of intersection: Set the function equal to 0 and solve for x.
• 2) Integrate the function: Use the definite integral notation and the points of intersection as the limits.

The Basics of Definite Integrals – Understand How to Calculate the Area Under a Curve

When working with functions, it is often necessary to find the area enclosed above and below the x-axis. Definite integrals provide a convenient method for accurately calculating this area with the help of definite integral notation and the points of intersection.

Introduction to Definite Integrals

Definite integrals are a mathematical tool used to determine the area between a continuous function and the x-axis within a specific interval. This is expressed in the form of an integral with upper and lower limits, for example, ∫^b_a f(x) dx.

In contrast to indefinite integrals, which result in a general function, definite integrals provide a specific numeric value as their outcome. This makes them particularly useful for finding the total area under a graph.

An essential aspect of definite integrals is their ability to account for areas above and below the x-axis. In cases where the enclosed area falls below the x-axis, the resulting definite integral will be negative. However, if the area is above the x-axis, the definite integral will be positive.

How to Calculate a Definite Integral

To accurately calculate a definite integral, it is crucial to first ensure that the function being integrated is continuous within the specified interval. This means that there should be no gaps or jumps in the graph that would affect the integration process.

Once the continuity of the function is confirmed, the definite integral can be computed by finding the area enclosed between the curve and the x-axis within the given limits.

Can Definite Integrals Have Negative Values?

Yes, definite integrals can have negative values. This occurs when the area below the x-axis is greater than the area above it within the given interval, resulting in a negative difference.

Summary

In summary, definite integrals are a useful mathematical tool for determining the area under a curve bounded by the x-axis within a specific interval. They provide a specific numeric value and account for both positive and negative areas. Understanding how to calculate definite integrals is essential for various applications in mathematics and beyond.