Area Between Curves: Functions of x - AP Calculus BC

1. Identify the area formula for two curves $f(x)$, $g(x)$ where $f(x) \neq g(x)$ on $[a, b]$.

   $\text{integral from } a \text{ to } b \text{ of } |f(x) - g(x)| \, dx$. Absolute value ensures positive area when curves switch positions.

2. What is the integral for area: $f(x) = \text{sin}(x)$, $g(x) = 0$, on $[0, \frac{\text{pi}}{2}]$?

   $\text{integral from } 0 \text{ to } \frac{\text{pi}}{2} \text{ of } \text{sin}(x) \, dx$. Since $g(x) = 0$, the area is the integral of $\sin(x)$.

3. In which scenario is $g(x)$ subtracted from $f(x)$?

   When $f(x) \text{ is above } g(x)$. The upper curve minus the lower curve gives positive area.

4. Which step is first: setting up limits or finding intersections?

   Finding intersections. Intersection points determine the integration limits.

5. Determine the area between $y=\text{sin}(x)$ and $y=0$ from $x=0$ to $x=\frac{\text{pi}}{2}$.

   1. Area = $\int_0^{\pi/2} \sin(x) dx = [-\cos(x)]_0^{\pi/2} = 1$.

6. State the condition for $f(x)$ and $g(x)$ in $[a, b]$ to use area formula.

   $f(x) \text{ and } g(x)$ must be continuous on $[a, b]$. Ensures the area formula is valid throughout the interval.

7. For $y=\text{e}^x$ and $y=1$, which function is on top for $x > 0$?

   $y=\text{e}^x$. For $x > 0$, $e^x > 1$ since the exponential function grows rapidly.

8. Find area between $y=x^3$ and $y=x$ from $x=0$ to $x=1$.

   $\frac{1}{4}$. Area = $\int_0^1 (x - x^3) dx = [\frac{x^2}{2} - \frac{x^4}{4}]_0^1 = \frac{1}{4}$.

9. What is the area under $y=4x-x^2$ and above $y=0$ from $x=0$ to $x=4$?

   $\frac{32}{3}$. Area = $\int_0^4 (4x - x^2) dx = [2x^2 - \frac{x^3}{3}]_0^4 = \frac{32}{3}$.

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

    $\frac{9}{2}$. Area = $\int_0^3 (3x - x^2) dx = [\frac{3x^2}{2} - \frac{x^3}{3}]_0^3 = \frac{9}{2}$.

11. State the formula to find the area between $f(x)$ and $g(x)$ over $[a, b]$.

    $\text{Area} = \int_a^b |f(x) - g(x)| \, dx$. Absolute value ensures positive area regardless of curve order.

12. What is the integral for area: $f(x) = \text{ln}(x)$, $g(x) = 0$, on $[1, \text{e}]$?

    $\text{integral from } 1 \text{ to } \text{e} \text{ of } \text{ln}(x) \, dx$. Since $g(x) = 0$, the area is the integral of $\ln(x)$.

13. Which curve is on top in the area formula if $f(x) > g(x)$ on $[a, b]$?

    $f(x)$ is on top. The curve with larger $y$-values is the upper curve.

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

    $\frac{1}{6}$. Area = $\int_0^1 (x - x^2) dx = [\frac{x^2}{2} - \frac{x^3}{3}]_0^1 = \frac{1}{6}$.

15. Identify the top curve: $y=\text{ln}(x)$ or $y=0$, for $x>1$.

    $y=\text{ln}(x)$. For $x > 1$, $\ln(x) > 0$, so it's above the $x$-axis.

16. Identify the curve to be subtracted when finding area: $y=f(x)$ above $y=g(x)$.

    Subtract $g(x)$ from $f(x)$. The lower curve is subtracted from the upper curve.

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

    $\text{Area} = \text{integral from } a \text{ to } b \text{ of } (f(x) - g(x)) \, dx$. Integrates the difference of the upper and lower functions.

18. What is the integral form for area: $f(x) = x^2$, $g(x) = 0$, between $x=0$ and $x=3$?

    $\int_0^3 x^2 \, dx$. Since $g(x) = 0$, the area is just $\int f(x) dx$.

19. Find the area between $y=\frac{1}{x}$ and $y=0$ from $x=1$ to $x=3$.

    $\text{ln}(3)$. Area = $\int_1^3 \frac{1}{x} dx = [\ln(x)]_1^3 = \ln(3)$.

20. What is the difference $f(x) - g(x)$ if $f(x)=3x$ and $g(x)=x^2$?

    $3x - x^2$. Subtracting the functions gives the integrand for area calculation.

21. Find area between $y=x$ and $y=x^3$ from $x=0$ to $x=1$.

    $\frac{1}{2}$. Area = $\int_0^1 (x - x^3) dx = [\frac{x^2}{2} - \frac{x^4}{4}]_0^1 = \frac{1}{4}$.

22. Find the area under $y=x^3$ and above $y=0$ from $x=0$ to $x=1$.

    $\frac{1}{4}$. Area = $\int_0^1 x^3 dx = [\frac{x^4}{4}]_0^1 = \frac{1}{4}$.

23. What is the integral setup for area: curves $y=\text{cos}(x)$, $y=\text{sin}(x)$, $x$ from $0$ to $\frac{\text{pi}}{4}$?

    $\text{integral from } 0 \text{ to } \frac{\text{pi}}{4} \text{ of } (\text{cos}(x) - \text{sin}(x)) \, dx$. For $x \in [0, \frac{\pi}{4}]$, $\cos(x) \geq \sin(x)$.

24. Identify the area formula if $f(x)$ is below $g(x)$ on $[a, b]$.

    $\text{integral from } a \text{ to } b \text{ of } (g(x) - f(x)) \, dx$. When $g(x) > f(x)$, subtract $f(x)$ from $g(x)$ for positive area.

25. What is the integral setup for area: $f(x)=x^2$, $g(x)=x^4$, on $[0, 1]$?

    $\text{integral from } 0 \text{ to } 1 \text{ of } (x^2 - x^4) \, dx$. For $x \in [0,1]$, $x^2 \geq x^4$ since $x^2(1-x^2) \geq 0$.

26. Calculate area between $y=4-x^2$ and $y=0$ from $x=0$ to $x=2$.

    $\frac{16}{3}$. Area = $\int_0^2 (4-x^2) dx = [4x - \frac{x^3}{3}]_0^2 = \frac{16}{3}$.

27. What is the integral setup for area: $f(x) = \frac{1}{x}$, $g(x) = 0$, $x \text{ from } 1 \text{ to } 2$?

    $\text{integral from } 1 \text{ to } 2 \text{ of } \frac{1}{x} \, dx$. Since $g(x) = 0$, the integrand is just $f(x)$.

28. Determine the limits of integration for $y=x^2$ and $y=2x-x^2$.

    $x=0$ to $x=2$. Found by solving $x^2 = 2x - x^2$, giving $x = 0$ and $x = 2$.

29. Find the area under $y=x^2$ above $y=0$ from $x=0$ to $x=2$.

    $\frac{8}{3}$. Area = $\int_0^2 x^2 dx = [\frac{x^3}{3}]_0^2 = \frac{8}{3}$.

30. State the method to find points of intersection of $f(x)$ and $g(x)$.

    Set $f(x) = g(x)$ and solve for $x$. Setting functions equal finds where curves cross.

31. Find the area between $y=x$ and $y=0$ from $x=0$ to $x=1$.

    $\frac{1}{2}$. Area = $\int_0^1 x dx = [\frac{x^2}{2}]_0^1 = \frac{1}{2}$.

32. Which function represents the upper curve: $y=2x$ or $y=x^2$, on $[0, 1]$?

    $y=2x$. For $x \in [0,1]$, $2x \geq x^2$ since $2x - x^2 = x(2-x) \geq 0$.

Study these flashcards