Area Between Curves quiz #1 Flashcards
Area Between Curves quiz #1
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How do you find the area between two curves y = f(x) and y = g(x) over a given interval [a, b]?
Set up the definite integral from a to b of the top function minus the bottom function: ∫[a to b] (f(x) - g(x)) dx, where f(x) is above g(x) on the interval. What should you do if the bounds for the area between two curves are not given?
Find the intersection points of the two functions by setting f(x) = g(x) and solving for x. Use these x-values as the bounds for your definite integral. How do you determine which function is the 'top' and which is the 'bottom' when setting up the area integral between two curves?
For each x in the interval, the 'top' function is the one with the greater y-value, and the 'bottom' function has the lesser y-value. Use the graph or compare function values to decide. What should you do if the two curves intersect within the interval of integration, causing the top and bottom functions to switch?
Split the interval at the intersection point(s) and set up separate integrals for each subinterval, adjusting which function is on top and which is on bottom for each part. Why is it important to subtract the bottom function from the top function when finding the area between curves?
Subtracting the bottom function from the top function ensures that the integrand represents the vertical distance between the curves, giving the correct area. If the area between y = 4 - x^2 and y = 2x + 1 is to be found, and the bounds are not given, how do you determine the limits of integration?
Set 4 - x^2 = 2x + 1 and solve for x to find the intersection points, which become the limits of integration. How do you set up the area integral if the top and bottom functions switch at x = c within [a, b]?
Set up two integrals: ∫[a to c] (top1(x) - bottom1(x)) dx + ∫[c to b] (top2(x) - bottom2(x)) dx, where the top and bottom functions are chosen appropriately for each interval. Does the method for finding the area between curves change if the curves are below the x-axis?
No, the method remains the same. The area is always found by integrating the top function minus the bottom function, regardless of their position relative to the x-axis. What is the general formula for the area between two curves y = f(x) and y = g(x) from x = a to x = b?
The area is given by ∫[a to b] |f(x) - g(x)| dx, where the absolute value ensures the area is always positive. How do you find the area between two curves y = f(x) and y = g(x) over a given interval [a, b]?
Set up the definite integral from a to b of the top function minus the bottom function: ∫[a to b] (f(x) - g(x)) dx, where f(x) is above g(x) on the interval.
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