Accumulation Functions, Definite Intervals, Applied Contexts - AP Calculus BC

1. How does $\int_a^b f(x)\,dx$ change if $a$ and $b$ are swapped?

   It becomes $-\int_b^a f(x)\,dx$. Reversing limits changes sign.

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

   $\int_0^2 (x^2 + 3)\,dx = \frac{26}{3}$. Using $[\frac{x^3}{3} + 3x]_0^2$.

3. What does the definite integral $\int_a^b r(t)\,dt$ represent in economics?

   Total accumulated revenue over time from $a$ to $b$. Integrating revenue rate gives total revenue.

4. What does $\int_0^T f(t)\,dt$ represent in population models?

   Total population change over time $0$ to $T$. Integrating population growth rate.

5. What does the Fundamental Theorem of Calculus Part 2 state?

   If $f$ is continuous on $[a, b]$, $F(x) = \int_a^x f(t)\,dt$ is continuous and differentiable. Shows accumulation functions are differentiable.

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

   $\int_0^4 (x^3 - 2x)\,dx = 48$. Using $[\frac{x^4}{4} - x^2]_0^4 = 64 - 16$.

7. What is the geometric interpretation of $\int_a^b f(x)\,dx$ when $f(x) \geq 0$?

   It is the area under the curve $f(x)$ from $x = a$ to $x = b$. Region bounded by curve and x-axis.

8. Find $A(x)$ given $f(t) = e^t$ and $a = 0$.

   $A(x) = \int_0^x e^t\,dt = e^x - 1$. Using $[e^t]_0^x = e^x - e^0$.

9. Determine the value of $\int_1^3 x^2\,dx$.

   $\int_1^3 x^2\,dx = 26/3$. Using $[\frac{x^3}{3}]_1^3 = 9 - \frac{1}{3}$.

10. Evaluate $\int_0^\pi \sin x\,dx$.

    $\int_0^\pi \sin x\,dx = 2$. $[-\cos x]_0^\pi = -(-1) - (-1)$.

11. Evaluate the integral of a constant, $\int_a^b c\,dx$.

    $c(b-a)$, where $c$ is a constant. Constant times interval length.

12. Express $\int_a^b f(x)\,dx$ using the limit of sums.

    $\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$. Riemann sum definition of definite integral.

13. Identify the expression for the average value of $f(x)$ on $[a, b]$.

    The average value is $\frac{1}{b-a} \int_a^b f(x)\,dx$. Divides total area by interval width.

14. What does $\frac{d}{dx} \int_a^x f(t)\,dt$ equal?

    $f(x)$, according to the Fundamental Theorem of Calculus Part 2. Derivative of accumulation function equals integrand.

15. What is the formula for accumulation function $A(x)$?

    $A(x) = \int_a^x f(t)\,dt$ where $a$ is a constant and $f(t)$ is a given function. Accumulates $f(t)$ from fixed point $a$ to variable $x$.

16. What is the integral of $\int_a^b kf(x)\,dx$ where $k$ is constant?

    $k \int_a^b f(x)\,dx$. Constant factor property of integrals.

17. What is the Fundamental Theorem of Calculus Part 1?

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

18. Determine the integral $\int_0^3 4x\,dx$.

    $\int_0^3 4x\,dx = 18$. Using $[2x^2]_0^3 = 2 \cdot 9$.

19. Find the accumulation function $A(x)$ given $f(t) = 3t^2$ and $a = 1$.

    $A(x) = \int_1^x 3t^2\,dt = x^3 - 1$. Using FTC: $\int_1^x 3t^2\,dt = [t^3]_1^x$.

20. Evaluate $\int_1^2 \frac{1}{x}\,dx$.

    $\int_1^2 \frac{1}{x}\,dx = \ln 2$. Natural logarithm is antiderivative of $\frac{1}{x}$.

21. How do you calculate the total distance traveled from $t = a$ to $t = b$?

    $\int_a^b |v(t)|\,dt$ for velocity function $v(t)$.. Absolute value ensures positive distances.

22. Find the value of $\int_0^2 (2 + x)\,dx$.

    $\int_0^2 (2 + x)\,dx = 6$. Using $[2x + \frac{x^2}{2}]_0^2$.

23. Calculate $\int_0^2 (2x + 1)\,dx$.

    $\int_0^2 (2x + 1)\,dx = 6$. Antiderivative: $[x^2 + x]_0^2 = 4 + 2 = 6$.

24. What is the integral of a function over an interval with zero length?

    Zero, since $\int_a^a f(x)\,dx = 0$. Degenerate interval has no area.

25. What is the accumulation function for $f(t) = \cos t$ from $a = 0$?

    $A(x) = \int_0^x \cos t\,dt = \sin x$. Using $[\sin t]_0^x = \sin x - 0$.

26. Express the net change of a quantity using integrals.

    Net change is $\int_a^b F'(x)\,dx = F(b) - F(a)$. FTC relates rate of change to total change.

27. What is the result of $\int_0^1 5\,dx$?

    5, since the integral of a constant is the constant times the interval length. Constant $5$ over interval length $1$.

28. What is $\int_a^a f(x)\,dx$ equal to?

    Zero, since the interval has zero length. No area when endpoints are identical.

29. How do you interpret $\int_a^b v(t)\,dt$ in a physics context?

    It represents the displacement of an object from time $a$ to $b$ given velocity $v(t)$.. Integrating velocity gives position change.

30. State the integration property for $\int_a^b [f(x) + g(x)]\,dx$.

    $\int_a^b f(x)\,dx + \int_a^b g(x)\,dx$. Linearity property of integrals.

31. Find $\int_0^1 e^x\,dx$.

    $\int_0^1 e^x\,dx = e - 1$. Using $[e^x]_0^1 = e^1 - e^0$.

32. Determine $\int_{-1}^1 x^3\,dx$.

    $\int_{-1}^1 x^3\,dx = 0$. Odd function over symmetric interval cancels.

33. State the relationship between definite integral and area under a curve.

    The definite integral $\int_a^b f(x)\,dx$ represents the net area between $f(x)$ and the x-axis. Positive areas above axis, negative below.

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