Calculus I: Applications of Integrals, Techniques of Integration, and Limits

Applications of Definite Integrals

Area Between Curves

The area between two curves can be found using definite integrals by integrating the difference of the functions over the interval where they intersect.

• Formula: $A = \int_{a}^{b} [f(x) - g(x)]\,dx$ where $f(x)$ is the upper function and $g(x)$ is the lower function on $[a, b]$.

• Example: Find the area between $y = x^2 + 2x + 1$ and $y = 2x + 1$ from $x = 0$ to $x = 1$: $A = \int_{0}^{1} [(2x + 1) - (x^2 + 2x + 1)]\,dx = \int_{0}^{1} -x^2\,dx$

Area Bounded by Trigonometric Functions

• When the curves involve trigonometric functions, set up the integral with the correct bounds and subtract the lower function from the upper function.

• Example: $A = \int_{0}^{\pi/2} [\sin(2x) - \sin(x)]\,dx$

Volume of Solids of Revolution

Volumes can be found by revolving a region around an axis using the disk, washer, or shell method.

• Disk Method: Used when the solid is generated by revolving a region around an axis and the cross-sections perpendicular to the axis are disks. $V = \pi \int_{a}^{b} [R(x)]^2 dx$

• Washer Method: Used when there is a hole in the middle (two curves). $V = \pi \int_{a}^{b} \left([R_{\text{outer}}(x)]^2 - [R_{\text{inner}}(x)]^2\right) dx$

• Shell Method: Used when integrating with respect to the axis perpendicular to the axis of revolution. $V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) dx$

• Example: Volume generated by revolving $y = x^2$ about the y-axis from $x = 0$ to $x = 2$: $V = 2\pi \int_{0}^{2} x(x^2) dx$

Techniques of Integration

Integration by Parts

Used for integrating products of functions.

• Formula: $\int u\,dv = uv - \int v\,du$

• Example: $\int x e^x dx$ Let $u = x$, $dv = e^x dx$; then $du = dx$, $v = e^x$. $\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C$

Integration of Trigonometric Functions

• Use identities and substitution to simplify integrals involving trigonometric functions.

• Example: $\int \sin^2(x) dx = \int \frac{1 - \cos(2x)}{2} dx$

Integration by Substitution

• Let $u = g(x)$, then $du = g'(x) dx$ and rewrite the integral in terms of $u$.

• Example: $\int x \cos(x^2) dx$ Let $u = x^2$, $du = 2x dx$, so $x dx = du/2$. $\int x \cos(x^2) dx = \frac{1}{2} \int \cos(u) du = \frac{1}{2} \sin(u) + C = \frac{1}{2} \sin(x^2) + C$

Integration of Exponential Functions

• Integrals involving $e^{ax}$ or $a^x$ can often be solved by substitution or by recognizing standard forms.

• Example: $\int e^{2x} dx = \frac{1}{2} e^{2x} + C$

Evaluating Limits

Limits Involving Indeterminate Forms

• Apply algebraic manipulation, L'Hôpital's Rule, or series expansion to evaluate limits.

• Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Limits at Infinity

• For rational functions, compare the degrees of the numerator and denominator.

• Example: $\lim_{x \to \infty} \frac{2x^2 + 3}{x^2 - 1} = 2$

Summary Table: Methods for Finding Area and Volume

Method	Formula	When to Use
Area between curves	$\int_{a}^{b} [f(x) - g(x)] dx$	Finding area between two curves
Disk Method	$\pi \int_{a}^{b} [R(x)]^2 dx$	Solid of revolution, no hole
Washer Method	$\pi \int_{a}^{b} ([R_{\text{outer}}(x)]^2 - [R_{\text{inner}}(x)]^2) dx$	Solid of revolution, with hole
Shell Method	$2\pi \int_{a}^{b} (\text{radius})(\text{height}) dx$	Solid of revolution, integrating perpendicular to axis

Additional info:

• Some problems require the use of trigonometric identities or integration by parts multiple times.

• Highlighted notes in the original file emphasize the importance of using long synthetic division for certain integrals and refer to trigonometric integration formulas from specific textbook sections.