Area Between Curves with Multiple Intersections - AP Calculus AB

1. What is the integral setup for the area between $y = 4x$ and $y = x^3$ on $[0, 2]$?

   $A = \int_0^2 (4x - x^3) \, dx$. $4x > x^3$ on this interval.

2. What is the formula for calculating area between curves if $f(x)$ and $g(x)$ switch?

   Separate integrals for each segment with correct top curve. Account for position changes at intersections.

3. Identify the integral for $y = x^2$ and $y = 5 - x^2$ on $[0, \sqrt{5}]$.

   $A = \int_0^{\sqrt{5}} (5 - 2x^2) \, dx$. $5 - x^2 > x^2$ gives $5 > 2x^2$.

4. Calculate the area between $y = 3x$ and $y = x^2$ from $x = 0$ to $x = 3$.

   $A = \int_0^3 (3x - x^2) \, dx$. $3x$ is above $x^2$ on this interval.

5. What is the area between $y = \sin(x)$ and $y = \cos(x)$ from $x = 0$ to $x = \frac{\pi}{4}$?

   $A = \int_0^{\frac{\pi}{4}} (\cos(x) - \sin(x)) \, dx$. $\cos(x) > \sin(x)$ on $[0, \frac{\pi}{4}]$.

6. What happens when $f(x)$ and $g(x)$ switch positions in the interval?

   Adjust the integrals to reflect the change in position. Swap the order in the integrand appropriately.

7. What do you do if curves intersect at more than two points?

   Split into separate integrals for each segment. Each segment needs its own integral calculation.

8. What is the area between $y = x^2$ and $y = 4x - x^2$ from $x = 0$ to $x = 2$?

   $A = \int_0^2 [(4x - x^2) - x^2] \, dx$. $4x - x^2$ is above $x^2$ on this interval.

9. What integral calculates the area between $y = x^2$ and $y = x$ on $[0, 1]$?

   $A = \int_0^1 (x - x^2) \, dx$. $y = x$ is above $y = x^2$ on $[0, 1]$.

10. Determine the intersection points of $y = x^3$ and $y = x$.

    Intersection points: $(0, 0)$ and $(1, 1)$. Solve $x^3 = x$ to find where curves meet.

11. State the process to find the area between curves crossing multiple times.

    Divide into segments, integrate each, sum the areas. Handle each interval with proper curve ordering.

12. Calculate the area between $y = x^3$ and $y = 2x^2$ from $x = 0$ to $x = 2$.

    $A = \int_0^2 (2x^2 - x^3) \, dx$. $2x^2 > x^3$ on this interval.

13. How is the area between $y = x^2$ and $y = x^3$ over $[0, 1]$ calculated?

    $A = \int_0^1 (x^2 - x^3) \, dx$. $x^2 > x^3$ for $x \in (0, 1)$.

14. Find the intersection points of $y = x$ and $y = x^2$.

    Intersection points: $(0, 0)$ and $(1, 1)$. Set $x = x^2$ and solve for intersection points.

15. For $y = 2x$ and $y = x^3$, what is the area on $[0, 2]$?

    $A = \int_0^2 (2x - x^3) \, dx$. $2x > x^3$ on this interval.

16. How do you handle regions where the curves intersect multiple times?

    Calculate separate integrals for each region. Different regions require separate area calculations.

17. Determine the area between $y = x^2$ and $y = 4 - x^2$ on $[-2, 2]$.

    $A = 2 \int_0^2 (4 - 2x^2) \, dx$. Uses symmetry since $4 - x^2 > x^2$ on $[-2, 2]$.

18. Find the area between $y = 2x$ and $y = x^2$ from $x = 0$ to $x = 2$.

    $A = \int_0^2 (2x - x^2) \, dx$. $2x$ is above $x^2$ on this interval.

19. What is the area between $y = 3x^2$ and $y = x^3$ from $x = 0$ to $x = 3$?

    $A = \int_0^3 (3x^2 - x^3) \, dx$. $3x^2 > x^3$ on this interval.

20. Write the integral for the area between $y = x$ and $y = x^3$ on $[0, 1]$.

    $A = \int_0^1 (x - x^3) \, dx$. $y = x$ is above $y = x^3$ on $[0, 1]$.

21. How do you verify which curve is on top in an interval?

    Evaluate the functions at a test point within the interval. Compare function values at any interior point.

22. What is the integral setup for $y = 4x - x^2$ and $y = x^2$ on $[0, 2]$?

    $A = \int_0^2 (4x - 2x^2) \, dx$. $4x - x^2$ is above $x^2$ on this interval.

23. What step determines the integration limits for the area between curves?

    Find the points of intersection of the curves. Intersections define where curves switch position.

24. For $y = x^2$ and $y = 2x$, what is the area on $[0, 2]$?

    $A = \int_0^2 (2x - x^2) \, dx$. $2x > x^2$ on this interval.

25. When should you re-evaluate curve positions in an area problem?

    If the curves intersect within the interval. Intersections change which curve is on top.

26. What is the general formula for the area between two curves $f(x)$ and $g(x)$?

    $A = \int_a^b [f(x) - g(x)] \, dx$. Subtracts lower curve from upper curve over the interval.

27. Identify the upper curve for $y = x^2$ and $y = 3 - x^2$ on $[0, 2]$.

    $y = 3 - x^2$ is the upper curve. $3 - x^2$ has larger values on this interval.

28. What is the area between $y = e^x$ and $y = 1$ from $x = 0$ to $x = 1$?

    $A = \int_0^1 (e^x - 1) \, dx$. $e^x > 1$ for all $x > 0$.

29. What integral calculates the area between $y = 4x$ and $y = x^2$ on $[0, 4]$?

    $A = \int_0^4 (4x - x^2) \, dx$. $4x > x^2$ on this interval.

30. What is the next step after finding intersections for area calculation?

    Determine which curve is above the other in each interval. Test points reveal which function is greater.

31. What are often the limits of integration for area problems?

    The x-coordinates of the intersections of the curves. Intersections naturally define integration bounds.

32. Which method can be used if curves intersect more than twice?

    Split the integration into regions based on intersections. Each segment has different upper/lower curves.

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