Applying Properties of Definite Integrals - AP Calculus BC

1. Evaluate $\int_0^4 (x + 1)\,dx$ using properties.

   12. $\int_0^4 x dx + \int_0^4 1 dx = 8 + 4 = 12$

2. What property allows combining integrals over adjacent intervals?

   Additivity. Allows splitting intervals at any point.

3. Evaluate $\int_0^{\pi} \sin(x)\,dx$ using known values.

   2. $[-\cos(x)]_0^\pi = -(-1) - (-1) = 2$

4. Evaluate $\int_0^3 (x^2 - x)\,dx$ using linearity.

   $\frac{9}{2}$. $\int_0^3 x^2 dx - \int_0^3 x dx = 9 - \frac{9}{2} = \frac{9}{2}$

5. Find the definite integral: $\int_0^2 3\,dx$.

   6. $3(2-0) = 6$

6. What is a definite integral's value if $f(x) = 0$ for all $x$?

   0. Zero function has no area.

7. Identify the property used: $\int_a^b (f(x) + g(x))\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx$.

   Linearity. Distributes integration over addition.

8. Evaluate $\int_0^1 x^3\,dx$ using known antiderivatives.

   $\frac{1}{4}$. $\left[\frac{x^4}{4}\right]_0^1 = \frac{1}{4}$

9. Evaluate $\int_2^5 4\,dx$ using properties.

   12. $4(5-2) = 12$

10. State the property of symmetry for odd functions in definite integrals.

    $\int_{-a}^a f(x)\,dx = 0$ if $f(x)$ is odd. Odd function symmetry cancels positive and negative areas.

11. Using properties, find $\int_0^3 (x^2 + 4x)\,dx$.

    27. $\int_0^3 x^2 dx + \int_0^3 4x dx = 9 + 18 = 27$

12. What is the definite integral of a zero function over any interval?

    0. Zero function integrates to zero.

13. What does the comparison property of definite integrals state?

    If $f(x) \leq g(x)$, then $\int_a^b f(x)\,dx \leq \int_a^b g(x)\,dx$. Preserves inequalities in integration.

14. Identify the property of definite integrals for an interval of zero length.

    $\int_a^a f(x)\,dx = 0$. Zero width interval gives zero area.

15. Evaluate $\int_{-2}^2 x^3\,dx$ using symmetry.

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

16. What is the property of additivity for definite integrals?

    $\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$. Combines adjacent intervals into one integral.

17. What is the definite integral of an odd function over symmetric limits?

    0. Odd functions over symmetric intervals cancel.

18. What is the property of definite integrals when reversing limits?

    $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$. Swapping limits changes sign.

19. How does the zero integral property affect definite integrals?

    $\int_a^a f(x)\,dx = 0$. No area when upper and lower limits are equal.

20. What is the property of symmetry for even functions in definite integrals?

    $\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx$ if $f(x)$ is even. Even function symmetry doubles half-interval.

21. Identify the integral property: $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$.

    Reversal of Limits. Changes sign when limits are swapped.

22. Express the linearity property of definite integrals.

    $\int_a^b (f(x) + g(x))\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx$. Integral of sum equals sum of integrals.

23. What is the formula for the definite integral of a constant function $c$?

    $\int_a^b c\,dx = c(b-a)$. Constant times interval width.

24. Evaluate $\int_1^4 5x^2\,dx$ using constant multiple property.

    105. $5 \cdot \int_1^4 x^2 dx = 5 \cdot 21 = 105$

25. State the property of definite integrals involving constant multiplication.

    $\int_a^b c\cdot f(x)\,dx = c\cdot \int_a^b f(x)\,dx$. Constants factor out of integrals.

26. What is $\frac{d}{dx} \int_a^x f(t)\,dt$?

    $f(x)$. Fundamental Theorem of Calculus, first part.

27. Find the derivative: $\frac{d}{dx} \int_2^x \sin(t)\,dt$.

    $\sin(x)$. By Fundamental Theorem of Calculus.

28. What is the effect of integration limits on the definite integral?

    $\int_a^b f(x)\,dx$ depends only on $f(x)$ between $a$ and $b$. Values outside limits don't affect the integral.

29. What is the integral property that allows splitting at a point?

    $\int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx$. Breaks integral at intermediate point.

30. What does the Mean Value Theorem for definite integrals state?

    There exists $c \in [a, b]$ such that $f(c) = \frac{1}{b-a}\int_a^b f(x)\,dx$. Guarantees average value exists somewhere in interval.

31. What is the integral of $\int_0^a cx\,dx$ where $c$ is constant?

    $\frac{ca^2}{2}$. Constant $c$ factors out, then integrate $x$.

32. What is $\frac{d}{dx} \int_0^x e^t\,dt$?

    $e^x$. By Fundamental Theorem of Calculus.

33. Find $\int_{-3}^3 (x^4 + x^2)\,dx$ using symmetry.

    72. Both functions are even, so use symmetry: $2\int_0^3 (x^4 + x^2)dx$

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