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# Integrals ## Lagrange Multipliers A method of optimization to find maximum/minimum values of $f(x, y, z)$ subject to a constraint $g(x, y, z) = 0$. We compute $\nabla f$ and $\nabla g$ and solve $\nabla f = \lambda \nabla g$ alongside $g(x, y, z) = 0$. Use solutions to solve as long as it exis...

# Integrals ## Lagrange Multipliers A method of optimization to find maximum/minimum values of $f(x, y, z)$ subject to a constraint $g(x, y, z) = 0$. We compute $\nabla f$ and $\nabla g$ and solve $\nabla f = \lambda \nabla g$ alongside $g(x, y, z) = 0$. Use solutions to solve as long as it exists. ## Double Integrals **Rectangular Region:** Computes the volume under a surface $z = f(x, y)$ over a rectangle $R = [a, b] \times [c, d]$. $$ \iint_R f(x, y) dA = \int_a^b \int_c^d f(x, y) \, dy \, dx $$ **Average Value:** $\frac{1}{\text{Area}(R)} \iint_R f(x, y) \, dA$ **General Region:** Usually boundaries include functions of the same sort. When $R: a \le x \le b, g_1(x) \le y \le g_2(x)$, $$ \iint_R f(x, y) dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \, dx $$ When $R: c \le y \le d, h_1(y) \le x \le h_2(y)$, $$ \iint_R f(x, y) dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x, y) \, dx \, dy $$ **Polar Coordinates:** $\qquad x = r \, \cos(\theta)$, $\qquad y = r \, \sin(\theta)$, $dA = r \, dr \, d\theta$. $$\iint_R f(x, y) dA = \int_{\theta_1}^{\theta_2} \int_{r_1}^{r_2} f( r \cos(\theta), r \sin(\theta) ) r \, dr \, d\theta$$ $$Volume = \iint_R f(x,y) \, dA$$ When converting from rectangular to polar, you just use $r$ and how much of the circle you want for $\theta$. ## Triple Integrals **General Type:** Finds the volume of solids or integrates functions over 3D regions. $$ \iiint_V f(x, y, z) \, dV $$ **Cylindrical Coordinates:** $x = r \cos(\theta)$, $y = r \sin(\theta)$, $z = z$, $dV = r \, dr \, d\theta \, dz$. Useful for solids with rotational symmetry around the z-axis. **Spherical Coordinates:** $x = \rho \sin(\phi) \cos(\theta)$, $y = \rho \sin(\phi) \sin(\theta)$, $z = \rho \cos(\phi)$, $dV = \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta$. Useful for spheres and hemispherical solids. The bounds for $\rho$ are the radius of the solid, the bounds for $\phi$ are the same as z (how much of the region from the positive z-axis), and the bounds for $\theta$ are how much rotation.