Fundamental Theorem of Calculus: Definite Intervals - AP Calculus BC

1. Calculate the integral of $f(x) = x^{-1}$ from 1 to 4.

   $\text{ln}(4)$. Same as $\int \frac{1}{x} dx$, antiderivative is $\ln(x)$.

2. Calculate the integral of $f(x) = \frac{1}{x^2}$ from 1 to 2.

   $\frac{1}{2}$. Antiderivative is $-\frac{1}{x}$, then $-\frac{1}{2} - (-1) = \frac{1}{2}$.

3. Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.

   $\frac{13}{4}$. Evaluate $x^4 - \frac{x^3}{3} + 2x$ at bounds 1 and 0.

4. What is the antiderivative of $f(x) = 2x^3$?

   $\frac{x^4}{2} + C$. Power rule integration: increase exponent, divide by new exponent.

5. Find the integral of $f(x) = 1 + x$ from 0 to 2.

   4. Antiderivative is $x + \frac{x^2}{2}$, evaluate at bounds.

6. Identify the antiderivative given $f(x) = 2x$.

   $F(x) = x^2 + C$. The antiderivative of $2x$ is $x^2$ plus a constant.

7. Determine the integral of $f(x) = x^4$ from 0 to 1.

   $\frac{1}{5}$. Antiderivative is $\frac{x^5}{5}$, then $\frac{1}{5} - 0$.

8. Find the integral of $f(x) = 3x^2 + 2x$ from 0 to 2.

   16. Evaluate $x^3 + x^2$ at bounds 2 and 0.

9. Evaluate the integral of $f(x) = x^3$ from 0 to 2.

   4. Antiderivative is $\frac{x^4}{4}$, then $\frac{16}{4} - 0 = 4$.

10. Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.

    $\text{ln}(2)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(2) - \ln(1)$.

11. What does the Fundamental Theorem of Calculus connect?

    Differentiation and integration. Links the inverse operations of calculus.

12. What is the integral of $f(x) = e^{2x}$ from 0 to 1?

    $\frac{e^2 - 1}{2}$. Antiderivative is $\frac{e^{2x}}{2}$, evaluate at bounds.

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

    15. Integral of constant function equals constant times interval length.

14. Evaluate the integral of $f(x) = x^3 - 3x^2 + 2x$ from 0 to 1.

    0. Evaluate $\frac{x^4}{4} - x^3 + x^2$ at bounds 1 and 0.

15. Determine the integral of $f(x) = e^x$ from 0 to 1.

    $e - 1$. Antiderivative of $e^x$ is $e^x$, then $e - 1$.

16. Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$.

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

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

    $b - a$. Antiderivative of 1 is $x$, difference of bounds gives length.

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

    16. Antiderivative is $2x^2$, then $18 - 2 = 16$.

19. Find the integral of $f(x) = 2x^2 - 3x + 1$ from 1 to 2.

    $\frac{7}{3}$. Evaluate $\frac{2x^3}{3} - \frac{3x^2}{2} + x$ at bounds.

20. Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.

    $\frac{9}{2}$. Evaluate $\frac{x^3}{3} - \frac{x^2}{2}$ at bounds 3 and 0.

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