1:33:46
Calculus 1 Lecture 5.1: Finding Area Between Two Curves
Professor Leonard
Overview
This video explains how to calculate the area between two curves using definite integrals. It builds upon the concept of finding the area under a single curve by introducing the idea of subtracting the area under the lower curve from the area under the upper curve. The video covers scenarios where the interval is given and where it needs to be determined by finding intersection points. It also touches upon applications in real-world problems, such as calculating the distance between two moving objects, and explores integrating with respect to y as an alternative method when functions are not easily expressed in terms of x.
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Chapters
- The area between two curves, f(x) and g(x), over an interval [a, b] can be found by integrating the difference between the upper curve and the lower curve.
- This is achieved by calculating the definite integral of the upper function and subtracting the definite integral of the lower function over the same interval.
- Mathematically, Area = ∫[a,b] (f(x) - g(x)) dx, where f(x) ≥ g(x) on [a, b].
- The resulting area will always be positive; a negative result indicates an error in setup, likely by reversing the upper and lower functions.
Understanding this concept allows you to calculate the precise area of regions bounded by functions, which is fundamental in many applications like physics and engineering.
If f(x) is above g(x) on the interval [a, b], the area between them is the area under f(x) minus the area under g(x).
- Finding the area under a single curve f(x) from a to b is a special case of finding the area between two curves.
- In this case, the second curve is the x-axis, represented by the function g(x) = 0.
- Therefore, the integral ∫[a,b] f(x) dx is equivalent to ∫[a,b] (f(x) - 0) dx.
- This explains why areas below the x-axis were previously treated as negative, as they represented the 'lower' function being subtracted from zero.
Recognizing this connection reinforces the understanding of how previous calculus concepts are extended and provides a unified framework for area calculations.
The area under f(x) from a to b is the same as the area between f(x) and the x-axis (y=0).
- When the interval [a, b] and the upper/lower functions are explicitly provided, the setup is straightforward.
- Identify the function that is consistently above the other within the given interval.
- Set up the integral as the difference between the upper function and the lower function, ensuring correct use of parentheses for subtraction.
- Evaluate the definite integral using standard integration techniques.
This is the most direct application of the area between curves formula, providing a foundational skill for more complex problems.
Find the area bounded above by 2x + 5 and below by x³ on the interval [0, 2]. The integral would be ∫[0,2] ((2x + 5) - x³) dx.
- If the interval is not given, it must be found by determining where the curves intersect.
- Set the two functions equal to each other (f(x) = g(x)) and solve for x to find the points of intersection.
- These intersection points serve as the bounds of integration (a and b).
- After finding the intersection points, test a value within the interval to determine which function is on top.
This skill is crucial for finding the area of enclosed regions where the boundaries are defined solely by the curves themselves.
To find the area between y = x² and y = x + 6, set x² = x + 6, solve to find intersections at x = -2 and x = 3, which become the integration bounds.
- If curves intersect at more than two points, the region may be divided into multiple sub-regions.
- Each sub-region will require a separate integral, as the function on top might change at each intersection point.
- The total area is the sum of the areas of these sub-regions.
- When setting up integrals, always ensure the function on top is listed first in the difference.
This addresses more complex bounded regions where a single integral is insufficient, requiring a piecewise approach to accurately sum all enclosed areas.
For y = x³ and y = x, intersections are at x = -1, 0, and 1. This creates two intervals: [-1, 0] and [0, 1], each requiring a separate integral based on which function is on top in that specific interval.
- The area between velocity curves represents the distance between the objects.
- Integrating the difference between two velocity functions over an interval gives the net distance gained by one object over the other.
- Some problems involve curves that are easier to express as x in terms of y (x = f(y)) rather than y in terms of x (y = f(x)).
- In such cases, it's more efficient to integrate with respect to y, using the rightmost curve minus the leftmost curve.
This demonstrates the practical relevance of area calculations in physics and introduces a powerful alternative integration strategy for complex curve geometries.
When comparing two cars' velocities, the area between their velocity-time graphs represents the difference in distance traveled by each car.
Key takeaways
- The area between two curves f(x) and g(x) over an interval [a, b] is found by integrating the difference between the upper and lower functions: ∫[a,b] (upper(x) - lower(x)) dx.
- The x-axis (y=0) acts as the 'lower' curve when calculating the area under a single function.
- If the integration interval is not given, find it by setting the two functions equal to each other and solving for x.
- When curves intersect multiple times, the area must be calculated as a sum of integrals over sub-intervals, with the top and bottom functions potentially switching.
- The area between velocity-time curves represents the difference in distance traveled by the objects.
- For curves not easily expressed as y = f(x), integrating with respect to y (using x = f(y)) can be a simpler approach.
- Always ensure the function on top is subtracted from the function on the bottom to get a positive area.
Key terms
Area between curvesDefinite integralUpper functionLower functionInterval of integrationIntersection pointsBounds of integrationNet areaTotal areaIntegrate with respect to y
Test your understanding
- How do you determine which function is the 'upper' function and which is the 'lower' function when calculating the area between two curves?
- What is the relationship between finding the area under a single curve and finding the area between two curves?
- Describe the process for finding the limits of integration when the interval is not explicitly provided.
- Why might it be necessary to split a region into multiple integrals when finding the area between two curves?
- How can the concept of integrating with respect to y simplify finding the area between certain curves?