Fundamental Theorem of Calculus: Definite Intervals - AP Calculus AB

1. Determine $\int_1^4 \frac{1}{x}\,dx$.

   $\ln(4)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$.

2. Evaluate $\int_0^4 (3x^2 + 2)\,dx$ using the Fundamental Theorem.

   68. Use antiderivative $x^3 + 2x$, evaluate at bounds $4$ and $0$.

3. For $F(x) = \int_0^x \,\sin(t)\,dt$, find $F'(x)$.

   $\sin(x)$. By Part 2 of FTC, derivative equals the integrand.

4. What is the result of $\int_0^2 1\,dx$?

   2. Integral of constant $1$ over interval length $2$.

5. What is a definite integral's geometric interpretation?

   Net area between the curve and the x-axis over $[a, b]$. Represents the signed area between curve and x-axis.

6. For $F(x) = \int_0^x e^t\,dt$, find $F'(x)$.

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

7. What is the integral of $f(x) = 1/x$ over $[1, e]$?

   1. Antiderivative of $\frac{1}{x}$ is $\ln(x)$; $\ln(e) - \ln(1) = 1$.

8. What condition must a function $f$ satisfy for the Fundamental Theorem to apply?

   $f$ must be continuous on $[a, b]$. Continuity ensures the antiderivative exists and is differentiable.

9. Find $\int_0^4 (x^2 - 2)\,dx$ using the Fundamental Theorem.

   $\frac{32}{3}$. Use antiderivative $\frac{x^3}{3} - 2x$, evaluate at bounds.

10. Define a proper antiderivative.

    A function $F(x)$ whose derivative is $f(x)$. An antiderivative is any function whose derivative gives $f(x)$.

11. Evaluate $\int_0^3 (5x - 4)\,dx$ using the Fundamental Theorem.

    15. Use antiderivative $\frac{5x^2}{2} - 4x$, evaluate at bounds.

12. Determine $\int_0^2 x^2\,dx$ using an antiderivative.

    $\frac{8}{3}$. Use antiderivative $\frac{x^3}{3}$, evaluate at bounds.

13. State the Fundamental Theorem of Calculus, Part 1.

    If $F$ is an antiderivative of $f$ on $[a,b]$, then $\int_a^b f(x)\,dx = F(b) - F(a)$. This allows calculation of definite integrals using antiderivatives.

14. Evaluate $\int_{-2}^2 (x^3 + 2x)\,dx$ using symmetry properties.

    $0$. Both functions are odd, so integral over symmetric interval is zero.

15. Find the derivative of $F(x) = \int_2^x \ln(t)\,dt$.

    $\ln(x)$. By Part 2 of FTC, the derivative equals the integrand.

16. Compute $\int_{-1}^1 x^3\,dx$ using symmetry properties.

    0. Odd function over symmetric interval gives zero area.

17. What is an antiderivative of $f(x) = 3x^2$?

    $F(x) = x^3 + C$. Power rule: increase exponent by 1, divide by new exponent.

18. Find the value of $\int_0^\pi \cos(x)\,dx$.

    0. Antiderivative is $\sin(x)$; $\sin(\pi) - \sin(0) = 0$.

19. Calculate $\int_0^1 (x^3 + 1)\,dx$ using the Fundamental Theorem.

    $\frac{5}{4}$. Use antiderivative $\frac{x^4}{4} + x$, evaluate at bounds.

20. What is the definite integral of a constant $c$ over an interval $[a, b]$?

    $c(b-a)$. A constant function integrates to constant times interval length.

21. Find $\int_0^\pi \sin(x)\,dx$ using the Fundamental Theorem.

    2. Antiderivative is $-\cos(x)$; $-\cos(\pi) - (-\cos(0)) = 2$.

22. Evaluate $\int_0^3 (x^2 + x + 1)\,dx$ using the Fundamental Theorem.

    18. Use antiderivative $\frac{x^3}{3} + \frac{x^2}{2} + x$, evaluate at bounds.

23. Evaluate $\int_0^1 (4x^3 - x)\,dx$ using the Fundamental Theorem.

    $\frac{3}{4}$. Use antiderivative $x^4 - \frac{x^2}{2}$, evaluate at bounds.

24. Evaluate $\int_0^2 (3x^2 - 2x)\,dx$ using the Fundamental Theorem.

    4. Use antiderivative $x^3 - x^2$, evaluate at bounds.

25. What is the purpose of the definite integral?

    To find the net area under a curve. It represents the signed area between curve and x-axis.

26. What is the result of a definite integral when the upper and lower limits are equal?

    0. When limits are equal, the interval has zero length.

27. State the Fundamental Theorem of Calculus, Part 2.

    If $f$ is continuous on $[a,b]$, then $F(x) = \int_a^x f(t)\,dt$ is differentiable and $F'(x) = f(x)$. Shows that differentiation and integration are inverse operations.

28. Calculate $\int_1^3 (2x^2 - x)\,dx$ using the Fundamental Theorem.

    $\frac{26}{3}$. Use antiderivative $\frac{2x^3}{3} - \frac{x^2}{2}$, evaluate at bounds.

29. What is the integral of $f(x) = e^x$ over $[0, 1]$?

    $e - 1$. Antiderivative of $e^x$ is $e^x$; evaluate at bounds.

30. What does the second part of the Fundamental Theorem allow us to compute?

    The derivative of an integral function. It gives the rate of change of the area function.

31. What is the integral of $f(x) = \sin(x)$ over $[0, \pi]$?

    2. Antiderivative is $-\cos(x)$; $-\cos(\pi) - (-\cos(0)) = 2$.

32. What does the Fundamental Theorem of Calculus connect?

    It connects differentiation and integration. They are inverse operations of each other.

33. What is the integral of $f(x) = \cos(x)$ over $[0, \pi]$?

    0. Antiderivative is $\sin(x)$; $\sin(\pi) - \sin(0) = 0$.

34. Find $\int_0^1 (x^2 + 2x)\,dx$ using an antiderivative.

    $\frac{7}{3}$. Use antiderivative $\frac{x^3}{3} + x^2$, evaluate at bounds.

35. Find $\int_0^2 (4x^3 + x)\,dx$ using an antiderivative.

    18. Use antiderivative $x^4 + \frac{x^2}{2}$, evaluate at bounds.

36. Determine $\int_1^3 x^2\,dx$ using an antiderivative.

    $26/3$. Use antiderivative $\frac{x^3}{3}$, then evaluate at bounds.

37. Determine $\int_0^1 (5x^2 - 3x)\,dx$ using an antiderivative.

    $\frac{1}{3}$. Use antiderivative $\frac{5x^3}{3} - \frac{3x^2}{2}$, evaluate at bounds.

38. Evaluate $\int_0^1 2x\,dx$ using the Fundamental Theorem of Calculus.

    1. Find antiderivative $x^2$, then evaluate $1^2 - 0^2 = 1$.

39. What does it mean for a function $F(x)$ to be differentiable?

    $F(x)$ has a derivative at every point in its domain. The function has a well-defined derivative at each point.

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