Exploring Accumulations of Change - AP Calculus AB

1. What is the integral of $f(x) = e^x$ from $0$ to $1$?

   $\int_0^1 e^x\,dx = e - 1$. Antiderivative is $e^x$, evaluated from 0 to 1 gives $e-1$.

2. What is the integral of $f(x) = 1$ from $0$ to $5$?

   $\int_0^5 1\,dx = 5$. Integral of constant function equals constant times interval length.

3. State the definition of an antiderivative.

   A function $F(x)$ such that $F'(x) = f(x)$ for all $x$ in the domain. Function whose derivative equals the given function.

4. Evaluate $\int_1^2 (x^2 + x)\,dx$.

   $\frac{7}{3}$. Sum of power functions integrated using power rule.

5. State the formula for the area under a curve using integration.

   Area = $\int_a^b f(x)\,dx$. Fundamental connection between integration and area under curves.

6. Compute $\int_0^4 (x - 2)\,dx$.

   $0$. Linear function with zero net area due to symmetry about $x=2$.

7. Find $\frac{d}{dx} \int_0^x \ln(t)\,dt$.

   $\ln(x)$. By FTC Part 1, derivative of integral equals integrand.

8. What is the integral of $f(x) = \sin(x)$ from $0$ to $\pi$?

   $\int_0^{\pi} \sin(x)\,dx = 2$. Antiderivative is $-\cos(x)$, giving total area under sine curve.

9. Evaluate $\int_0^{\pi} \sin(x)\,dx$.

   $2$. Antiderivative is $-\cos(x)$, evaluated from 0 to $\pi$ gives $-(-1)-(-1)=2$.

10. State the definition of a definite integral.

    The limit of Riemann sums: $\lim_{n\to\infty} \sum_{i=1}^{n} f(x_i^*)\Delta x$. Formal definition using limit of approximating rectangular areas.

11. What is the integral of $f(x) = 3x^2$ from $1$ to $4$?

    $\int_1^4 3x^2\,dx = 63$. Antiderivative is $x^3$, so $F(4) - F(1) = 64 - 1 = 63$.

12. Evaluate $\int_0^1 (x^4 - x^2 + 1)\,dx$.

    $\frac{5}{6}$. Sum of power functions integrated using standard power rule.

13. Find $\frac{d}{dx} \int_0^x e^{t^2}\,dt$.

    $e^{x^2}$. By FTC Part 1, derivative equals the integrand with $x$ substituted.

14. Find $\frac{d}{dx} \int_0^x \sin(t)\,dt$.

    $\sin(x)$. By FTC Part 1, derivative of integral with variable upper limit equals integrand.

15. What is the integral of $f(x) = 4x^3$ from $0$ to $1$?

    $\int_0^1 4x^3\,dx = 1$. Power rule: antiderivative of $x^3$ is $\frac{x^4}{4}$.

16. Compute $\int_0^2 (4x - x^2)\,dx$.

    $\frac{16}{3}$. Quadratic function forming parabolic region with positive area.

17. What is the integral of $f(x) = 2x$ from $1$ to $3$?

    $\int_1^3 2x\,dx = 8$. Antiderivative is $x^2$, evaluated from 1 to 3 gives $9-1=8$.

18. What is the average value of $f(x) = x^2$ on $[0, 3]$?

    $\frac{1}{3} \int_0^3 x^2\,dx = 3$. Average value formula: $\frac{1}{b-a}\int_a^b f(x)dx$ applied to $x^2$ on $[0,3]$.

19. Find the antiderivative of $f(x) = 5x^4$.

    $F(x) = x^5 + C$. Power rule for antiderivatives: increase exponent by 1, divide by new exponent.

20. What is the integral of $f(x) = \cos(x)$ from $0$ to $\frac{\pi}{2}$?

    $\int_0^{\frac{\pi}{2}} \cos(x)\,dx = 1$. Antiderivative is $\sin(x)$, evaluated from 0 to $\frac{\pi}{2}$ gives $1-0=1$.

21. What is the integral of $f(x) = 3$ from $0$ to $2$?

    $\int_0^2 3\,dx = 6$. Integral of constant equals constant times interval width.

22. Compute $\int_0^1 (x^2 - x + 1)\,dx$.

    $\frac{5}{6}$. Quadratic function integrated using power rule for each term.

23. Find $\int_0^2 (x^3 - 3x^2 + 3x)\,dx$.

    $2$. Cubic polynomial integrated using power rule for each term.

24. Compute $\int_0^2 (3x^2 + 2x)\,dx$.

    $12$. Antiderivative is $x^3 + x^2$, evaluated from 0 to 2 gives $8 + 4 = 12$.

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

    $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F$ is an antiderivative of $f$. States that definite integral equals antiderivative evaluated at bounds.

26. What is the integral of $f(x) = 5x$ from $0$ to $3$?

    $\int_0^3 5x\,dx = 22.5$. Linear function $5x$ has antiderivative $\frac{5x^2}{2}$.

27. What is the integral of $f(x) = x^3$ from $-1$ to $1$?

    $0$. Odd function over symmetric interval gives zero due to cancellation.

28. State the Mean Value Theorem for Integrals.

    $\exists c \in [a, b]$ such that $f(c) = \frac{1}{b-a}\int_a^b f(x)\,dx$. Guarantees existence of point where function equals its average value.

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