Integration applications take calculus beyond theoretical concepts into real-world problem-solving. We'll explore how to calculate areas between curves, volumes of solids, and physical quantities like work and force using definite integrals.

We'll also dive into more advanced applications like arc length and surface area of revolution. These techniques showcase the power of integration in modeling complex shapes and solving practical engineering problems.

Area Between Curves

Setting Up Definite Integrals

• Calculate the area between two curves by setting up a definite integral with the difference between the upper and lower functions as the integrand
• When the curves intersect, split the area into regions and add the definite integrals for each region together
• Use absolute value signs in the integrand to ensure a positive area if the upper and lower functions switch positions (e.g., $\int_{a}^{b} |f(x) - g(x)| dx$)
• Find horizontally oriented regions by integrating with respect to y instead of x, using the inverse functions (e.g., $\int_{c}^{d} |f^{-1}(y) - g^{-1}(y)| dy$)

Examples and Applications

• Find the area between the curves $y = x^2$ and $y = x + 2$ over the interval $[0, 2]$
• Calculate the area enclosed by the curves $y = \sin(x)$ and $y = \cos(x)$ from $x = 0$ to $x = \pi/2$
• Determine the area of the region bounded by the curves $y = e^x$ and $y = e^{-x}$ and the lines $x = -1$ and $x = 1$
• Find the area between the curves $x = y^2$ and $x = y + 1$ using integration with respect to y

Volume of Solids

Disk Method

• Calculate volumes of solids of revolution by integrating the area of circular cross-sections perpendicular to the axis of revolution
• The area of each circular cross-section is $\pi r^2$, where r is a function of x or y depending on the axis of revolution
• For a solid formed by revolving the region between $y = f(x)$ and $y = g(x)$ around the x-axis, the volume is given by $V = \int_{a}^{b} \pi[f(x)^2 - g(x)^2]dx$
• For a solid formed by revolving the region between $x = f(y)$ and $x = g(y)$ around the y-axis, the volume is given by $V = \int_{c}^{d} \pi[f(y)^2 - g(y)^2]dy$

Shell Method

• Calculate volumes of solids of revolution by integrating the volume of cylindrical shells parallel to the axis of revolution
• The volume of each cylindrical shell is $2\pi rh$, where r is the distance from the shell to the axis of revolution and h is the height of the shell
• For a solid formed by revolving the region between $y = f(x)$ and $y = g(x)$ around the y-axis, the volume is given by $V = \int_{a}^{b} 2\pi x[f(x) - g(x)]dx$
• For a solid formed by revolving the region between $x = f(y)$ and $x = g(y)$ around the x-axis, the volume is given by $V = \int_{c}^{d} 2\pi y[f(y) - g(y)]dy$
• Choose between the disk and shell methods based on the shape of the solid and which method simplifies the integration

Examples and Applications

• Find the volume of the solid formed by revolving the region bounded by $y = x^2$ and $y = 4$ around the x-axis using the disk method
• Calculate the volume of the solid formed by revolving the region bounded by $y = \sqrt{x}$, $x = 0$, and $y = 2$ around the y-axis using the shell method
• Determine the volume of the solid formed by revolving the region bounded by $y = \sin(x)$ and the x-axis from $x = 0$ to $x = \pi$ around the x-axis
• Find the volume of the solid formed by revolving the region bounded by $x = y^2 - 1$, $x = 0$, and $y = 2$ around the y-axis

Applications of Integration

Physical Quantities

• Use integration to calculate physical quantities such as work, force, pressure, mass, and center of mass
• Work done by a variable force $F(x)$ over a distance from a to b is calculated by the definite integral $W = \int_{a}^{b} F(x)dx$
• Hydrostatic force on a vertical surface submerged in a liquid is calculated by integrating the pressure (which varies with depth) over the submerged area (e.g., $F = \int_{0}^{h} \rho gA(y)dy$)
• Mass of an object with variable density $\rho(x, y, z)$ is calculated by integrating the density over the volume of the object (e.g., $m = \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} \rho(x, y, z)dxdydz$)
• Center of mass coordinates are calculated by dividing the integrals of x, y, or z times the density by the total mass (e.g., $\bar{x} = \frac{\int_{a}^{b} x\rho(x)dx}{\int_{a}^{b} \rho(x)dx}$)

Examples and Applications

• Calculate the work done by a spring with force $F(x) = kx$ as it is compressed from its natural length to a distance of 0.2 meters
• Find the hydrostatic force on a triangular dam with a base of 30 meters and a height of 20 meters when the water level is at the top of the dam
• Determine the mass of a cone with a radius of 5 cm and a height of 10 cm if its density varies linearly from 2 g/cm³ at the base to 1 g/cm³ at the vertex
• Calculate the center of mass of a thin wire with length L and density $\rho(x) = x^2$ from $x = 0$ to $x = L$

Arc Length and Surface Area

Arc Length

• Arc length is the distance along a curve, calculated by integrating the square root of $1 + (dy/dx)^2$ or $1 + (dx/dy)^2$ depending on the orientation
• For a curve given by $y = f(x)$ from $x = a$ to $x = b$, the arc length is given by $L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2}dx$
• For a curve given by $x = g(y)$ from $y = c$ to $y = d$, the arc length is given by $L = \int_{c}^{d} \sqrt{1 + [g'(y)]^2}dy$
• Parametric curves can also be used to calculate arc length by integrating with respect to the parameter (e.g., $L = \int_{t_1}^{t_2} \sqrt{[x'(t)]^2 + [y'(t)]^2}dt$)

Surface Area of Revolution

• Surface area of a solid of revolution is calculated by integrating the surface area of each infinitesimal strip formed by revolving the curve
• For curves rotated around the x-axis, the surface area integral is $2\pi$ times the integral of y times the square root of $1 + (dy/dx)^2$ (i.e., $S = 2\pi \int_{a}^{b} y\sqrt{1 + [f'(x)]^2}dx$)
• For curves rotated around the y-axis, the surface area integral is $2\pi$ times the integral of x times the square root of $1 + (dx/dy)^2$ (i.e., $S = 2\pi \int_{c}^{d} x\sqrt{1 + [g'(y)]^2}dy$)
• Parametric curves can also be used to calculate surface area by integrating with respect to the parameter (e.g., $S = 2\pi \int_{t_1}^{t_2} y(t)\sqrt{[x'(t)]^2 + [y'(t)]^2}dt$)

Examples and Applications

• Find the arc length of the curve $y = \ln(x)$ from $x = 1$ to $x = e$
• Calculate the arc length of the parametric curve $x = \cos(t)$, $y = \sin(t)$ from $t = 0$ to $t = 2\pi$
• Determine the surface area of the solid formed by revolving the curve $y = \sqrt{x}$ from $x = 0$ to $x = 4$ around the x-axis
• Find the surface area of the solid formed by revolving the parametric curve $x = \cos(t)$, $y = \sin(t)$ from $t = 0$ to $t = \pi$ around the y-axis