Integral Calculus

Cheatsheet Content

### Improper Integrals Improper integrals are definite integrals that have either infinite limits of integration or an integrand that becomes infinite at one or more points within the interval of integration. #### Type 1: Infinite Limits - $\int_a^\infty f(x) dx = \lim_{b \to \infty} \int_a^b f(x) dx$ - $\int_{-\infty}^b f(x) dx = \lim_{a \to -\infty} \int_a^b f(x) dx$ - $\int_{-\infty}^\infty f(x) dx = \int_{-\infty}^c f(x) dx + \int_c^\infty f(x) dx$ (for any real c) #### Type 2: Discontinuous Integrand If $f(x)$ is discontinuous at $x=a$ (and continuous on $(a,b]$): - $\int_a^b f(x) dx = \lim_{t \to a^+} \int_t^b f(x) dx$ If $f(x)$ is discontinuous at $x=b$ (and continuous on $[a,b)$): - $\int_a^b f(x) dx = \lim_{t \to b^-} \int_a^t f(x) dx$ If $f(x)$ is discontinuous at $x=c$ where $a 1$ and diverges if $p \le 1$. - **p-integral (Type 2):** $\int_0^1 \frac{1}{x^p} dx$ converges if $p ### Beta and Gamma Functions #### Gamma Function ($\Gamma(z)$) - **Definition:** $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} dt$, for $\text{Re}(z) > 0$. - **Recursive Property:** $\Gamma(z+1) = z\Gamma(z)$. - **Factorial Relation:** For positive integer $n$, $\Gamma(n+1) = n!$. - **Special Values:** - $\Gamma(1) = 1$ - $\Gamma(1/2) = \sqrt{\pi}$ - **Reflection Formula:** $\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$ - **Duplication Formula (Legendre's Formula):** $\Gamma(z)\Gamma(z+1/2) = 2^{1-2z}\sqrt{\pi}\Gamma(2z)$ #### Beta Function ($B(x,y)$) - **Definition:** $B(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt$, for $\text{Re}(x) > 0, \text{Re}(y) > 0$. - **Symmetry:** $B(x,y) = B(y,x)$. - **Relation to Gamma Function:** $B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$. - **Alternative Forms:** - $B(x,y) = 2\int_0^{\pi/2} \sin^{2x-1}\theta \cos^{2y-1}\theta d\theta$ - $B(x,y) = \int_0^\infty \frac{t^{x-1}}{(1+t)^{x+y}} dt$ ### Differentiation Under Integral Sign (Leibniz Integral Rule) If $F(t) = \int_{a(t)}^{b(t)} f(x,t) dx$, then: $$ \frac{dF}{dt} = f(b(t), t) \frac{db}{dt} - f(a(t), t) \frac{da}{dt} + \int_{a(t)}^{b(t)} \frac{\partial}{\partial t} f(x,t) dx $$ If the limits of integration are constants, i.e., $a(t) = a$ and $b(t) = b$: $$ \frac{dF}{dt} = \int_a^b \frac{\partial}{\partial t} f(x,t) dx $$ This rule is useful for evaluating certain definite integrals by differentiating with respect to a parameter. ### Double and Triple Integrals #### Double Integrals - **Definition:** For a function $f(x,y)$ over a region $R$ in the $xy$-plane: $$ \iint_R f(x,y) dA $$ - **Iterated Integrals:** - If $R = \{(x,y) | a \le x \le b, g_1(x) \le y \le g_2(x)\}$, then $\iint_R f(x,y) dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) dy dx$. - If $R = \{(x,y) | c \le y \le d, h_1(y) \le x \le h_2(y)\}$, then $\iint_R f(x,y) dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y) dx dy$. - **Properties:** Linearity, additivity over regions. #### Triple Integrals - **Definition:** For a function $f(x,y,z)$ over a solid region $E$ in 3D space: $$ \iiint_E f(x,y,z) dV $$ - **Iterated Integrals:** Similar to double integrals, extending to three variables. Example over a rectangular box $[a,b] \times [c,d] \times [e,f]$: $$ \int_a^b \int_c^d \int_e^f f(x,y,z) dz dy dx $$ - **Properties:** Linearity, additivity over regions. ### Areas and Volumes #### Area using Double Integrals The area of a plane region $R$ is given by: $$ \text{Area}(R) = \iint_R 1 dA $$ #### Volume using Double Integrals The volume of the solid under the surface $z=f(x,y)$ and above the region $R$ in the $xy$-plane is: $$ \text{Volume} = \iint_R f(x,y) dA $$ (assuming $f(x,y) \ge 0$) #### Volume using Triple Integrals The volume of a solid region $E$ is given by: $$ \text{Volume}(E) = \iiint_E 1 dV $$ ### Change of Order of Integration Changing the order of integration for iterated integrals is often necessary when the original order is difficult or impossible to evaluate. This involves sketching the region of integration to determine the new limits. **Example:** Original: $\int_0^1 \int_y^1 f(x,y) dx dy$ Region: $y \le x \le 1$, $0 \le y \le 1$. This is a triangle with vertices (0,0), (1,0), (1,1). New order (integrating with respect to $y$ first): $0 \le y \le x$, $0 \le x \le 1$. Result: $\int_0^1 \int_0^x f(x,y) dy dx$ ### Change of Variables in Multiple Integrals #### Double Integrals If we transform from $(x,y)$ to $(u,v)$ coordinates, where $x=g(u,v)$ and $y=h(u,v)$: $$ \iint_R f(x,y) dA = \iint_S f(g(u,v), h(u,v)) \left| \frac{\partial(x,y)}{\partial(u,v)} \right| du dv $$ The **Jacobian determinant** is given by: $$ J = \frac{\partial(x,y)}{\partial(u,v)} = \det \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} = \frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial y}{\partial u} $$ - **Polar Coordinates:** $x = r\cos\theta$, $y = r\sin\theta$. - Jacobian: $\left| \frac{\partial(x,y)}{\partial(r,\theta)} \right| = r$. - $dA = dx dy = r dr d\theta$. #### Triple Integrals If we transform from $(x,y,z)$ to $(u,v,w)$ coordinates: $$ \iiint_E f(x,y,z) dV = \iiint_F f(g(u,v,w), h(u,v,w), k(u,v,w)) \left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right| du dv dw $$ The **Jacobian determinant** is: $$ J = \frac{\partial(x,y,z)}{\partial(u,v,w)} = \det \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{pmatrix} $$ - **Cylindrical Coordinates:** $x = r\cos\theta$, $y = r\sin\theta$, $z = z$. - Jacobian: $\left| \frac{\partial(x,y,z)}{\partial(r,\theta,z)} \right| = r$. - $dV = dx dy dz = r dr d\theta dz$. - **Spherical Coordinates:** $x = \rho\sin\phi\cos\theta$, $y = \rho\sin\phi\sin\theta$, $z = \rho\cos\phi$. - Jacobian: $\left| \frac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)} \right| = \rho^2\sin\phi$. - $dV = dx dy dz = \rho^2\sin\phi d\rho d\phi d\theta$.