1. Integrate: Find the anti-derivative ($F$). Leave off the $+C$.
2. Upper Bound: Plug the top number ($b$) into $F(x)$.
3. Lower Bound: Plug the bottom number ($a$) into $F(x)$.
4. Subtract: Result $= F(b) - F(a)$.
Use the FTC to find the exact area for each function.
$\int_{0}^{3} x^2 dx$
$\int_{1}^{4} (2x + 5) dx$
$\int_{0}^{\pi} \sin x dx$
Why is the area under a Sine wave from 0 to $\pi$ exactly 2? Try to visualize the wave. Does it seem strange that a "Curved" wave gives such a clean "Whole" number for its area? What does this tell you about the design of the circle?
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$\int_{0}^{2} e^x dx$
$\int_{1}^{e} \frac{1}{x} dx$
Use the formula $\text{Avg} = \frac{1}{b-a} \int_{a}^{b} f(x) dx$.
Find the average value of $f(x) = x^2$ on the interval $[0, 3]$.
A person's "Growth Rate" is given by $v(t) = 3t^2 - 12t$.
1. Calculate $\int_{0}^{4} v(t) dt$.
2. Notice your answer is zero.
3. Does this mean they didn't do anything? Or does it mean they returned to the same spot?
Objective: Explain the FTC to a younger student using a ruler.
The Activity:
1. Lay a ruler on the table.
2. Place a toy at 2 inches.
3. Move the toy to 8 inches.
4. "I didn't count every tiny sliver of distance. I just subtracted 2 from 8."
The Lesson: "Math can find the 'Total' by just looking at the 'Before' and 'After'."
Response: ___________________________________________________________