Fundamental Theorem of Calculus: Accumulation Functions - AP Calculus AB

1. What does the Fundamental Theorem of Calculus connect?

   Differentiation and integration. The FTC bridges these two fundamental operations.

2. Find $\int_{-2}^{2} x^3 \, dx$.

   0. $x^3$ is odd, so integral over symmetric interval is zero.

3. What is the Fundamental Theorem of Calculus, Part 1?

   If $F$ is antiderivative of $f$, then $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$. This connects antiderivatives with definite integrals.

4. What is the value of $\int_{-a}^{a} f(x) \, dx$ if $f$ is an odd function?

   0. Odd functions have symmetric areas that cancel over symmetric intervals.

5. State the antiderivative of $f(x) = \frac{1}{x}$.

   $F(x) = \ln |x| + C$. Standard antiderivative formula for reciprocal function.

6. What is the result of $\int_{a}^{b} f(x) \, dx = 0$?

   The areas above and below the x-axis are equal. Net area is zero when positive and negative areas cancel.

7. What condition must a function $f$ meet to use the Fundamental Theorem of Calculus?

   $f$ must be continuous on $[a, b]$. Continuity ensures the theorem applies.

8. Find $\frac{d}{dx} \int_{2}^{x} (3t^2 + 2) \, dt$.

   $3x^2 + 2$. By FTC Part 2, derivative of integral equals the integrand.

9. Evaluate $\int_{1}^{4} \frac{1}{x} \, dx$.

   $\ln 4$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$; evaluate: $\ln(4) - \ln(1) = \ln(4)$.

10. State the relationship between definite integrals and net area.

    Definite integrals represent the net area under a curve. Positive area above x-axis minus negative area below.

11. What is $\int_{a}^{b} [f(x) + g(x)] \, dx$?

    $\int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx$. Linearity property of integration.

12. Evaluate $\int_{1}^{3} (2x+1) \, dx$.

    10. Antiderivative is $x^2 + x$; evaluate: $(9+3) - (1+1) = 10$.

13. What is the geometric interpretation of $\int_{a}^{b} f(x) \, dx$?

    Net area between $f(x)$ and the x-axis from $x=a$ to $x=b$. Signed area between curve and x-axis over the interval.

14. Find $\frac{d}{dx} \int_{a}^{g(x)} f(t) \, dt$ using the chain rule.

    $f(g(x))g'(x)$. Chain rule applied to variable upper limit of integration.

15. What is the Fundamental Theorem of Calculus, Part 2?

    If $f$ is continuous on $[a, b]$, then $g(x) = \int_{a}^{x} f(t) \, dt$ is differentiable and $g'(x) = f(x)$. Shows that differentiation undoes integration for accumulation functions.

16. Determine $\int_{1}^{2} (2x - 1) \, dx$ using an antiderivative.

    1. Antiderivative is $x^2 - x$; evaluate: $(4-2) - (1-1) = 1$.

17. What is the meaning of $\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx$?

    Additivity over intervals. Integrals can be split at any intermediate point.

18. Evaluate $\int_{0}^{\pi} \sin x \, dx$.

    2. Antiderivative of $\sin x$ is $-\cos x$; evaluate: $-\cos(\pi) - (-\cos(0)) = 2$.

19. What property does $\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$ illustrate?

    Reversal of limits changes the sign. Fundamental property showing orientation matters.

20. Find $\frac{d}{dx} \int_{0}^{x} e^t \, dt$.

    $e^x$. By FTC Part 2, derivative equals the integrand.

21. Evaluate $\int_{0}^{2} x \, dx$ using the Fundamental Theorem of Calculus.

    2. Antiderivative is $\frac{x^2}{2}$; evaluate: $\frac{4}{2} - 0 = 2$.

22. Identify the error: $\int_{a}^{b} f(x) \, dx = F(b) + F(a)$ where $F$ is an antiderivative of $f$.

    Correct: $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$. Should subtract, not add: $F(b) - F(a)$.

23. What is the integral of $e^x$ from $a$ to $b$?

    $\int_{a}^{b} e^x \, dx = e^b - e^a$. Standard exponential function integration formula.

24. State the effect of swapping integration limits on the integral value.

    It negates the integral. Swapping limits introduces a negative sign.

25. What does it imply if $\int_{a}^{b} f(x) \, dx > 0$?

    Net area above the x-axis is greater than below. Positive integral means more area above than below x-axis.

26. Evaluate $\int_{0}^{3} (4x + 1) \, dx$ using the Fundamental Theorem of Calculus.

    $\frac{45}{2}$. Find antiderivative $2x^2 + x$, evaluate at limits: $(18+3) - (0) = \frac{45}{2}$.

27. What is the integral of a constant $c$ from $a$ to $b$?

    $\int_{a}^{b} c \, dx = c(b-a)$. Constant times the width of the interval.

28. What does $\int_{a}^{a} f(x) \, dx$ equal for any function $f$?

    0. When integration limits are equal, the integral is zero.

29. State the formula for the derivative of an accumulation function $G(x) = \int_{a}^{x} f(t) \, dt$.

    $G'(x) = f(x)$ if $f$ is continuous. Direct application of FTC Part 2.

30. Evaluate $\int_{0}^{4} (3x^2) \, dx$ using the Fundamental Theorem of Calculus.

    64. Antiderivative is $x^3$; evaluate: $64 - 0 = 64$.

31. What is $\frac{d}{dx} \int_{x}^{1} t^2 \, dt$?

    $-x^2$. Swapping limits introduces negative sign by FTC.

32. What is the integral of $x^n$ from $a$ to $b$ for $n \neq -1$?

    $\int_{a}^{b} x^n \, dx = \frac{1}{n+1}(b^{n+1} - a^{n+1})$. Apply power rule for integration to definite integrals.

33. Determine $\int_{0}^{1} (2x^3 + 3x^2) \, dx$.

    $\frac{5}{2}$. Antiderivative is $\frac{x^4}{2} + x^3$; evaluate at limits: $\frac{5}{2}$.

Study these flashcards