Area Between Curves: Functions of x - AP Calculus AB

1. Find the intersection points of $y = x^2$ and $y = 4$.

   $x = -2, x = 2$. Solve $x^2 = 4$ to get $x = \pm 2$.

2. What is the integral for the area between $y = x^2$ and $y = -x$ from $x = 0$ to $x = 1$?

   $\int_{0}^{1} [x^2 - (-x)] \, dx$. Parabola is above line $y = -x$ on this interval.

3. Calculate the area between $y = x + 1$ and $y = 2x$ from $x = 0$ to $x = 1$.

   $\frac{1}{2}$. Compute $\int_0^1 [(x+1) - 2x]dx = \int_0^1 (1-x)dx$.

4. What is the first step to find the area between two curves $f(x)$ and $g(x)$?

   Identify intersection points of $f(x)$ and $g(x)$. Intersection points determine the limits of integration.

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

   $x = 0, x = 2$. Solve $2x = x^2$ to find where curves intersect.

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

   $\frac{32}{3}$. Evaluate $\int_{-2}^2 (4-x^2)dx$ using symmetry.

7. Identify the lower function for $f(x) = x^2 + 1$ and $g(x) = x + 2$ on $[0, 1]$.

   $f(x) = x^2 + 1$. Shifted parabola is below line on this interval.

8. Identify the upper function for $f(x) = x^3$ and $g(x) = x$ on $[-1, 1]$.

   $g(x) = x$. Line is above cubic on symmetric interval around origin.

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

   $\frac{8}{3}$. Line $y=2x$ is above parabola on $[0,2]$.

10. Determine the integral for the area between $y = x^2$ and $y = 4$ from $x = -2$ to $x = 2$.

    $\int_{-2}^{2} [4 - x^2] \, dx$. Horizontal line is above parabola from $-2$ to $2$.

11. Which function is lower: $y = x^3$ or $y = x^2$ on $[0, 1]$?

    $y = x^3$. Cubic grows slower than quadratic on $[0,1]$.

12. Determine the area between $y = x^3$ and $y = x$ from $x = 0$ to $x = 1$.

    $\frac{1}{4}$. Line $y=x$ is above cubic $y=x^3$ on $[0,1]$.

13. What is the integral to find the area between $y = x^2$ and $y = 4$ from $x = 0$ to $x = 2$?

    $\int_{0}^{2} [4 - x^2] \, dx$. Horizontal line is above parabola on this interval.

14. Which function is lower: $y = x^3$ or $y = x$ on $[-1, 0]$?

    $y = x^3$. Cubic function is negative while line is positive on $[-1,0]$.

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

    $f(x) \geq g(x)$ over interval $[a, b]$. Upper function must be greater than or equal to lower function.

16. How do you express the area between $y = 3x$ and $y = x^2$ on $[0, 3]$?

    $\int_{0}^{3} [3x - x^2] \, dx$. Line $y=3x$ is above parabola on $[0,3]$.

17. Calculate the area between $y = x^2 - 1$ and $y = 2x - 1$ from $x = 0$ to $x = 1$.

    $\frac{1}{6}$. Line $y=2x-1$ is above shifted parabola on $[0,1]$.

18. Identify the upper function for $f(x) = x^2 + 1$ and $g(x) = x^2$ on $[0, 1]$.

    $f(x) = x^2 + 1$. Shifted parabola is above original parabola.

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

    $\int_{0}^{3} [3x - x^2] \, dx$. Same setup as previous identical problem.

20. Identify the upper function for $f(x) = x^2$ and $g(x) = x + 2$ on $[0, 3]$.

    $g(x) = x + 2$. Linear function is above parabola in this interval.

21. Find the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$.

    $\frac{1}{6}$. Evaluate $\int_0^1 (x-x^2)dx$ for line above parabola.

22. What is the integral for the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$?

    $\int_{0}^{1} [x - x^2] \, dx$. Line is above parabola on unit interval.

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

    $\int_{0}^{3} [3x - x^2] \, dx$. Line is above parabola on the given interval.

24. What is the purpose of finding intersection points of $f(x)$ and $g(x)$?

    To determine limits of integration. Intersection points become the integration bounds.

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

    $\frac{8}{3}$. Same calculation as previous identical problem.

26. What is the area formula for two curves $f(x)$ and $g(x)$ over $[a, b]$?

    $\int_{a}^{b} |f(x) - g(x)| \, dx$. Absolute value ensures positive area regardless of function order.

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

    $\int_{a}^{b} [f(x) - g(x)] \, dx$. Integral of upper function minus lower function over the interval.

28. How do you find the limits of integration for two curves $f(x)$ and $g(x)$?

    Solve $f(x) = g(x)$. Set functions equal and solve for intersection points.

29. Identify the upper function for $f(x) = 2x$ and $g(x) = x^2$ on $[0, 2]$.

    $f(x) = 2x$. Line has greater slope than parabola on $[0,2]$.

30. Calculate the area between $y = x^2 + 1$ and $y = 1$ from $x = -1$ to $x = 1$.

    $\frac{8}{3}$. Same calculation as before for shifted parabola.

31. Identify the lower function for $f(x) = x^2$ and $g(x) = x + 2$ on $[2, 3]$.

    $f(x) = x^2$. Parabola is below linear function on this interval.

32. What does $\int_{a}^{b} [f(x) - g(x)] \, dx$ represent geometrically?

    Area between $f(x)$ and $g(x)$ on $[a, b]$. The definite integral represents signed area between curves.

33. What is the geometric interpretation of $\int_{a}^{b} [g(x) - f(x)] \, dx$?

    Area between $g(x)$ and $f(x)$ on $[a, b]$. When $g(x) \geq f(x)$, integral gives area between curves.

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