📐Analytic Geometry and Calculus Unit 9 – Integration Applications: Area & Volume

Key Concepts and Definitions

• Integration involves finding the area under a curve or the volume of a solid formed by rotating a curve around an axis
• Definite integrals calculate the area or volume within specific bounds (lower and upper limits)
• Indefinite integrals find the general antiderivative of a function without specific bounds
• Solids of revolution are 3D shapes formed by rotating a 2D curve around an axis (x-axis or y-axis)
• Disk method calculates the volume of a solid of revolution by approximating it with thin cylindrical disks
• Washer method calculates the volume of a solid of revolution by approximating it with thin cylindrical washers
• Shell method calculates the volume of a solid of revolution by approximating it with thin cylindrical shells
  • Useful when the curves are not easily integrated with respect to x or y

Integration Basics Review

• Integration is the reverse process of differentiation and is used to find the area under a curve or the volume of a solid
• The fundamental theorem of calculus connects differentiation and integration
  • It states that $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is an antiderivative of $f(x)$
• Indefinite integrals are written as $\int f(x) dx = F(x) + C$, where $C$ is a constant of integration
• Definite integrals have specific bounds and are written as $\int_a^b f(x) dx$, where $a$ and $b$ are the lower and upper limits
• Integration techniques include u-substitution, integration by parts, and trigonometric substitution
• The area under a curve can be approximated using Riemann sums, which divide the area into rectangles and sum their areas
• The limit of Riemann sums as the number of rectangles approaches infinity gives the exact area under the curve

Area Between Curves

• To find the area between two curves, integrate the difference of the upper and lower functions
• The area between curves $f(x)$ and $g(x)$ from $x=a$ to $x=b$ is given by $\int_a^b [f(x) - g(x)] dx$
  • Ensure that $f(x) \geq g(x)$ on the interval $[a,b]$
• If the curves intersect, split the integral at the point of intersection and calculate the areas separately
• When given curves in parametric form, convert them to functions of x or y before integrating
• Area between curves can be used to solve problems in physics, economics, and engineering (work done, consumer surplus)

Volume of Solids of Revolution

• Solids of revolution are formed by rotating a 2D curve around an axis (x-axis or y-axis)
• The volume of a solid of revolution can be calculated using the disk method, washer method, or shell method
• The choice of method depends on the shape of the solid and the axis of rotation
• Disk method approximates the solid with thin cylindrical disks perpendicular to the axis of rotation
  • Volume of each disk is $\pi r^2 dx$ or $\pi r^2 dy$, where $r$ is the radius of the disk
• Washer method approximates the solid with thin cylindrical washers perpendicular to the axis of rotation
  • Volume of each washer is $\pi (R^2 - r^2) dx$ or $\pi (R^2 - r^2) dy$, where $R$ and $r$ are the outer and inner radii of the washer
• Shell method approximates the solid with thin cylindrical shells parallel to the axis of rotation
  • Volume of each shell is $2\pi rh dr$, where $r$ is the distance from the axis of rotation to the shell and $h$ is the height of the shell

Methods for Finding Volumes

• Disk method: $V = \int_a^b \pi [f(x)]^2 dx$ (rotating around x-axis) or $V = \int_c^d \pi [g(y)]^2 dy$ (rotating around y-axis)
  • Used when the solid can be easily approximated by disks perpendicular to the axis of rotation
• Washer method: $V = \int_a^b \pi ([f(x)]^2 - [g(x)]^2) dx$ (rotating around x-axis) or $V = \int_c^d \pi ([f(y)]^2 - [g(y)]^2) dy$ (rotating around y-axis)
  • Used when the solid has a hole in the middle and can be approximated by washers perpendicular to the axis of rotation
• Shell method: $V = \int_a^b 2\pi xf(x) dx$ (rotating around y-axis) or $V = \int_c^d 2\pi yg(y) dy$ (rotating around x-axis)
  • Used when the solid can be easily approximated by shells parallel to the axis of rotation
• Choose the method that simplifies the integration process based on the given functions and the axis of rotation
• Sketch the region being rotated to visualize the solid and determine the appropriate method

Applications in Real-World Problems

• Volume of containers, tanks, and vessels can be calculated using integration (oil drums, water tanks)
• Work done by a variable force over a distance can be found by integrating the force function (spring compression, hydraulic systems)
• Centroid and center of mass of irregular shapes can be determined using integration (beams, plates)
• Fluid pressure and force on submerged surfaces can be calculated using integration (dams, underwater gates)
• Probability density functions in statistics can be integrated to find probabilities over specific intervals (normal distribution, exponential distribution)
• Electric and magnetic fields can be calculated using integration (charged plates, current-carrying wires)
• Flow rates and total flow in pipes and channels can be determined using integration (water supply systems, irrigation channels)

Common Mistakes and How to Avoid Them

• Forgetting to include the constant of integration (+ C) in indefinite integrals
  • Always include the constant of integration when finding the general antiderivative
• Incorrectly setting up the limits of integration
  • Carefully identify the bounds of the region being integrated and use them as the limits of integration
• Misidentifying the upper and lower functions when finding the area between curves
  • Ensure that the upper function is greater than or equal to the lower function on the given interval
• Using the wrong formula for the chosen volume method
  • Double-check the formula for the disk, washer, or shell method based on the axis of rotation and the given functions
• Incorrectly setting up the integral when using the shell method
  • Remember to multiply the integrand by $2\pi x$ or $2\pi y$ depending on the axis of rotation
• Forgetting to split the integral when the curves intersect
  • If the curves intersect, find the point(s) of intersection and split the integral into separate regions
• Misinterpreting the problem statement and using the wrong integration technique
  • Carefully read the problem and identify the key information needed to solve it using the appropriate integration method

Practice Problems and Solutions

1. Find the area between the curves $y = x^2$ and $y = x + 2$ from $x = -1$ to $x = 2$.
   • Solution: $\int_{-1}^2 [(x+2) - x^2] dx = \frac{9}{2}$
2. Calculate the volume of the solid formed by rotating the region bounded by $y = \sqrt{x}$, $y = 0$, $x = 0$, and $x = 4$ about the x-axis.
   • Solution: $V = \int_0^4 \pi (\sqrt{x})^2 dx = \frac{32\pi}{5}$
3. Find the volume of the solid formed by rotating the region bounded by $y = x^2$, $y = 4x$, and $x = 0$ about the y-axis using the shell method.
   • Solution: $V = \int_0^4 2\pi x(4x - x^2) dx = \frac{128\pi}{5}$
4. Determine the area between the curves $y = \sin x$ and $y = \cos x$ from $x = 0$ to $x = \frac{\pi}{2}$.
   • Solution: $\int_0^{\frac{\pi}{2}} [\sin x - \cos x] dx = 2$
5. A tank has a circular base with a radius of 2 meters and a height of 5 meters. If the tank is filled with water to a depth of 3 meters, find the volume of water in the tank.
   • Solution: $V = \int_0^3 \pi (2)^2 dy = 12\pi$ cubic meters
6. Find the centroid of the region bounded by the curve $y = x^3$ and the x-axis from $x = 0$ to $x = 2$.
   • Solution: $\bar{x} = \frac{\int_0^2 x(x^3) dx}{\int_0^2 x^3 dx} = \frac{8}{5}$, $\bar{y} = \frac{\int_0^2 \frac{1}{2}(x^3)^2 dx}{\int_0^2 x^3 dx} = \frac{16}{15}$
7. Calculate the work done by a force $F(x) = x^2 + 1$ in moving an object from $x = 0$ to $x = 4$.
   • Solution: $W = \int_0^4 (x^2 + 1) dx = \frac{80}{3}$ joules