Exam Cram Sheet: Calculus I Integration Mastery

This guide provides an ultra-condensed review of the fundamental theorems, techniques, and applications of integration required for the Calculus I curriculum.

Topic Weighting

Based on typical Calculus I curricula, expect the following distribution in a final exam:

Key Concepts Summary

1. The Fundamental Theorem of Calculus (FTC)

• Part 1 (The Derivative of an Integral): $\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$. This shows that differentiation and integration are inverse processes.
• Part 2 (Evaluation Theorem): $\int_{a}^{b} f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$.

2. $u$-Substitution

Used when the integrand contains a function and its derivative (Chain Rule in reverse).

• Choose $u = g(x)$ such that $du = g'(x)dx$ simplifies the integral.
• Crucial: Change the limits of integration for definite integrals immediately after choosing $u$.

3. Visualizing Volume Methods

Common Pitfalls

• The Missing $+C$: Always include the constant of integration for indefinite integrals.
• **Bounds in $u-Sub:** Forgetting to convert the x$-bounds to $u$-bounds. If $u=x^2$ and $x \in [1,2]$, then $u \in [1,4]$.
• Washer Order: Subtracting $(r_{inner} - r_{outer})^2$ instead of $(r_{outer}^2 - r_{inner}^2)$.
• Variable Mismatch: Integrating with respect to $x when the function is defined in terms of y$ (especially in area between curves).
• Average Value vs. Net Change: Average value requires dividing by the interval: $\frac{1}{b-a} \int_a^b f(x) dx$. Net change is just $\int_a^b f'(x) dx$.

Mnemonics / Memory Triggers

[!TIP] "S-C-U-B-A" for $u$-Substitution

1. Select $u$.
2. Calculate $du$.
3. Update bounds (for definite integrals).
4. Back-substitute $u$ and $du$ into the integral.
5. Antidifferentiate.

• **Integral of $\frac{1}{x}:** "Log is the natural choice for the inverse of x$." (Remember: $\int \frac{1}{x} dx = \ln|x| + C$).
• Disk vs. Shell: "Disks are Dead-on" (axis is perpendicular to the radius), while "Shells are Shadows" (parallel to the axis).

Formula / Equation Sheet

Basic Integrals

Applications

• Arc Length: $L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx$
• Work: $W = \int_{a}^{b} F(x) dx$
• Hydrostatic Force: $F = \int_{a}^{b} \rho g (depth) (width) dy$

Practice Set

1. Fundamental Theorem: Find $\frac{d}{dx} \int_{3}^{x^2} \sin(t) dt$.
   (Hint: Use the Chain Rule)
2. $u$-Substitution: Evaluate $\int x \sqrt{x^2 + 1} dx$.
3. Area Between Curves: Find the area bounded by $y = x^2$ and $y = x+2$.
4. Volume by Disk: Rotate the region under $y = \sqrt{x}$ from $x=0$ to $x=4$ around the $x$-axis.
5. Net Change: A particle has velocity $v(t) = 3t^2 - 6t. Find the total distance traveled between t=0$ and $t=3$.

▶Click for Solutions

1. Solution: $2x \cdot \sin(x^2)$ (Substitute $x^2$ and multiply by its derivative).
2. Solution: $\frac{1}{3}(x^2+1)^{3/2} + C$ (Let $u = x^2+1$).
3. Solution: $\int_{-1}^{2} (x+2 - x^2) dx = 4.5$.
4. Solution: $\pi \int_0^4 (\sqrt{x})^2 dx = \pi \int_0^4 x dx = 8\pi$.
5. Solution: $\int_0^2 |3t^2 - 6t| dt + \int_2^3 |3t^2 - 6t| dt$. Note the velocity changes sign at $t=2$.