Area Between Curves: Functions of y - AP Calculus AB

1. Convert $y = x^2$ and $x = 4$ to functions of $y$ for integration.

   $x = \text{sqrt}(y)$ and $x = 4$. Solve $y=x^2$ for $x$ to get $x=\sqrt{y}$; keep $x=4$.

2. Find points of intersection for $x = y^2$ and $x = 2 - y$.

   Solve $y^2 = 2 - y$. Set functions equal to find intersection $y$-coordinates.

3. What is the integral expression for area between $x = y$ and $x = y^2$?

   $\text{integral of } (y - y^2) \text{ dy}$. Since $y > y^2$ on $(0,1)$, integrate $(y-y^2)$.

4. What is the condition for $x = f(y)$ to be left of $x = g(y)$?

   $f(y) < g(y)$. For curves expressed as $x=f(y)$ and $x=g(y)$.

5. Identify $y$-limits for $x = y^2 + 1$ and $x = y + 3$.

   Solve $y^2 + 1 = y + 3$. Set functions equal to find $y$-intersection points.

6. Find area between $x = 4 - y^2$ and $x = y^2$ from $y = -2$ to $y = 2$.

   $\text{integral of } (4 - 2y^2) \text{ dy}$. Since $4-y^2 > y^2$ when $y^2 < 2$, integrate $(4-2y^2)$.

7. State the condition for $f(y)$ and $g(y)$ to find area between curves.

   $f(y) \text{ should be greater than } g(y) \text{ in the interval}$. Ensures $f(y) - g(y) \geq 0$ for positive area calculation.

8. What is the role of intersection points in finding areas?

   They determine limits of integration. Intersections provide the integration bounds for area calculation.

9. For curves $x = y^3$ and $x = 2y$, identify $y_1$ and $y_2$ if they intersect at $y = -1$ and $y = 2$.

   $y_1 = -1, y_2 = 2$. These are the $y$-coordinates where the curves intersect.

10. Find intersections of $x = 2y$ and $x = y^2$.

    Solve $2y = y^2$. Set the functions equal: $2y = y^2$ to find intersections.

11. Determine $f(y)$ and $g(y)$ for $x = y + 2$ and $x = 1$ for $y \text{ in } [0, 3]$.

    $f(y) = y + 2, g(y) = 1$. Identify which function is rightmost for each $y$-value.

12. State the integral for area between $x = y^2 + y$ and $x = y + 3$.

    $\text{integral of } (y + 3 - (y^2 + y)) \text{ dy}$. Simplifies to integral of $(3-y^2)$ after expanding.

13. What is $g(y)$ for the curve $x = y^2 + 1$?

    $g(y) = y^2 + 1$. Direct identification of the function from the equation.

14. Determine $f(y)$ and $g(y)$ for $x = y^2 + 1$ and $x = y + 3$.

    $f(y) = y + 3, g(y) = y^2 + 1$. Identify which function is rightmost to determine $f(y)$ and $g(y)$.

15. What is $f(y)$ for the curve $x = 1 - y^2$?

    $f(y) = 1 - y^2$. Direct identification of the function from the equation.

16. Find the limits of integration for $x = y + 1$ and $x = -y + 3$.

    Solve $y + 1 = -y + 3$. Set the functions equal to solve for intersection $y$-values.

17. Which curve is leftmost: $x = y^2$ or $x = y + 2$ for $y \text{ in } [0, 3]$?

    $x = y^2$. Compare function values: $y^2$ is smaller than $y+2$ on $[0,3]$.

18. Which function is rightmost: $x = y^2$ or $x = 3$?

    $x = 3$. Constant function $x=3$ is always to the right of $x=y^2$.

19. Calculate intersection points for $x = y^2$ and $x = 4 - y^2$.

    Solve $y^2 = 4 - y^2$. Set functions equal: $y^2 = 4-y^2$ gives $2y^2 = 4$.

20. State the formula for area between $x = f(y)$ and $x = g(y)$ using integration.

    $\text{Area} = \text{integral of } [f(y) - g(y)] \text{ dy}$. General area formula between two curves expressed as functions of $y$.

21. Which function is upper: $x = y + 1$ or $x = y^2$ for $y \text{ in } [0, 1]$?

    $x = y + 1$. For $y \in [0,1]$: $y+1 > y^2$ since line above parabola.

22. What is the formula for finding area between two curves $x = f(y)$ and $x = g(y)$?

    $\text{Area} = \int_{y_1}^{y_2} [f(y) - g(y)] \, \text{dy}$. Standard formula where $f(y) > g(y)$ for the entire interval.

23. Determine area between $x = y^2$ and $x = 4$ from $y = 0$ to $y = 2$.

    $\text{integral of } (4 - y^2) \text{ dy}$. Since $4 > y^2$ on $[0,2]$, integrate $(4-y^2)$.

24. Identify $y_1$ and $y_2$ for $x = 3y$ and $x = y^2$ if intersecting at $y = 0$ and $y = 3$.

    $y_1 = 0, y_2 = 3$. These are the given intersection $y$-coordinates for the curves.

25. Find the area between $x = y^2$ and $x = 4$ from $y = -2$ to $y = 2$.

    $\text{Area} = \text{integral from } -2 \text{ to } 2 \text{ of } (4 - y^2) \text{ dy}$. Since $4 > y^2$ for $y \in [-2,2]$, integrate $(4-y^2)$.

26. What must be true about limits $y_1$ and $y_2$ for integration?

    $y_1 < y_2$. Lower limit must be less than upper limit for proper integration.

27. Find the area between $x = 3y$ and $x = y^2$ from $y = 0$ to $y = 3$.

    $\text{Area} = \int_0^3 (3y - y^2) \, dy$. Since $3y > y^2$ on $[0,3]$, integrate $(3y-y^2)$.

28. Find the intersection points of $x = y^2 - 3$ and $x = 5 - y$.

    Solve $y^2 - 3 = 5 - y$. Set the functions equal to find intersection points.

29. Determine area between $x = 4$ and $x = y^2$ from $y = -2$ to $y = 2$.

    $\text{Area} = \text{integral from } -2 \text{ to } 2 \text{ of } (4 - y^2) \text{ dy}$. Same setup as flashcard 3: $4 > y^2$ on the interval.

30. Which function is upper: $x = y^2$ or $x = 4$ for $y \text{ in } [-2, 2]$?

    $x = 4$. Constant function $x=4$ is always to the right of $x=y^2$.

31. Identify the region of integration for curves $x = f(y)$ and $x = g(y)$ between $y = a$ and $y = b$.

    From $y = a$ to $y = b$. Integration bounds for functions of $y$ are the $y$-values.

32. What does the integral of $[f(y) - g(y)]$ represent when $x = f(y)$ and $x = g(y)$?

    Area between the curves. The definite integral gives the signed area between curves.

33. Find the intersection points for $x = 3y - y^2$ and $x = y$.

    Solve $3y - y^2 = y$. Set functions equal to find where curves intersect.

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