All Flashcards
Define 'area between curves'.
The area enclosed by two or more functions, found by integrating the difference between the functions over a given interval.
What is a definite integral?
A definite integral calculates the area under a curve between two specified limits (bounds) on the x-axis.
Define 'top function' in the context of area between curves.
The function with greater y-values over a given interval, used as the minuend in the integral.
Define 'bottom function' in the context of area between curves.
The function with lesser y-values over a given interval, used as the subtrahend in the integral.
What are the bounds of integration?
The x-values where the curves intersect, defining the interval over which the area is calculated.
Define intersection points of two curves.
Points where the graphs of two functions have the same x and y values.
What is the role of a graphing calculator in finding area between curves?
A tool to find intersection points and visualize which function is 'on top'.
What does 'enclosed region' mean?
The area bounded by the intersection of two or more curves.
Define the term 'integrand'.
The function that is being integrated.
What does it mean to 'evaluate a definite integral'?
To find the numerical value of the definite integral, representing the area.
Why do we subtract the integrals of two functions to find the area between them?
Subtracting the integrals effectively removes the area under the lower function from the area under the upper function, leaving only the area between them.
Explain the significance of finding intersection points when calculating the area between curves.
Intersection points define the limits of integration, indicating where the curves bound the area.
Why is it important to determine which function is 'on top'?
The 'top' function must be the minuend to ensure the area is positive, as area is always a positive quantity.
What happens if you integrate with the incorrect order (bottom - top)?
The result will be the negative of the actual area. Take the absolute value to correct.
How does the definite integral relate to the area?
The definite integral gives the signed area between a curve and the x-axis. For area between curves, it gives the area between the two curves.
Explain why absolute value is sometimes needed when finding the area between curves.
Absolute value is needed when the top and bottom functions switch places within the interval of integration, ensuring the area is always positive.
Concept of finding area between curves.
The area between two curves is found by integrating the difference of the top and bottom functions over an interval [a, b].
What is the geometric interpretation of the definite integral?
It represents the area under a curve between two points on the x-axis.
What happens if the functions intersect multiple times?
You must divide the integral into multiple integrals, one for each region where the top and bottom functions remain the same.
Explain the importance of the Fundamental Theorem of Calculus.
It links differentiation and integration, allowing us to evaluate definite integrals using antiderivatives.
Steps to find area between curves.
- Find intersection points. 2. Determine top/bottom functions. 3. Set up the integral: . 4. Evaluate the integral.
How to find intersection points?
- Set . 2. Solve for x. 3. Use a calculator if needed.
How to determine top/bottom function?
- Graph the functions. 2. Choose a test point within the interval. 3. Evaluate both functions at the test point; the larger value is the 'top' function.
How to set up the definite integral?
- Identify the limits of integration (a and b). 2. Write the integrand as (top function - bottom function). 3. Include 'dx'.
How to evaluate the definite integral?
- Find the antiderivative of the integrand. 2. Evaluate the antiderivative at the upper and lower limits. 3. Subtract: F(b) - F(a).
Steps if the top/bottom functions switch.
- Find all intersection points. 2. Split the integral into multiple integrals at each intersection. 3. Determine the top/bottom function for each interval. 4. Sum the absolute values of each integral.
How to check your answer?
- Graph the functions and visually estimate the area. 2. Compare your calculated area with the estimate. 3. Use a calculator to verify the definite integral.
What to do if you can't find the antiderivative?
Use a calculator with numerical integration capabilities to approximate the definite integral.
Steps to solve area between curves FRQ.
- Find intersection points using calculator. 2. Set up integral. 3. Evaluate integral using calculator.
How to find area between and ?
- Find intersection points. 2. Determine which function is on top. 3. Integrate the difference between the functions.