♾️AP Calculus AB/BC Unit 8 – Applications of Integration

Key Concepts

• Integration involves finding the area under a curve, which represents the accumulation of a quantity over an interval
• The definite integral $\int_a^b f(x) dx$ represents the area under the curve $f(x)$ from $x=a$ to $x=b$
• Antiderivatives are functions whose derivative is the original function, and indefinite integrals represent a family of antiderivatives
• Integration techniques include substitution, integration by parts, partial fractions, and trigonometric substitution
• Applications of integration include finding areas, volumes, arc lengths, surface areas, work, fluid pressure, center of mass, and moments
• The Fundamental Theorem of Calculus connects differentiation and integration, allowing for the evaluation of definite integrals using antiderivatives
• Riemann sums approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles
  • Left Riemann sums use the left endpoint of each subinterval to determine the rectangle heights
  • Right Riemann sums use the right endpoint of each subinterval to determine the rectangle heights
  • Midpoint Riemann sums use the midpoint of each subinterval to determine the rectangle heights

Fundamental Theorem of Calculus

• The Fundamental Theorem of Calculus (FTC) establishes the relationship between differentiation and integration
• The First Fundamental Theorem of Calculus states that if $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$, then $\int_a^b f(x) dx = F(b) - F(a)$
  • This theorem allows for the evaluation of definite integrals using antiderivatives
• The Second Fundamental Theorem of Calculus states that if $f$ is continuous on $[a,b]$, then $\frac{d}{dx} \int_a^x f(t) dt = f(x)$
  • This theorem relates the derivative of an integral to the original function
• The FTC enables the computation of definite integrals without using Riemann sums or limit processes
• The Mean Value Theorem for Integrals is a consequence of the FTC and states that for a continuous function $f$ on $[a,b]$, there exists a point $c$ in $[a,b]$ such that $\int_a^b f(x) dx = f(c)(b-a)$
• The FTC is used in various applications, such as finding the area under a curve, the volume of solids, and the work done by a variable force

Area Between Curves

• The area between two curves can be found by integrating the difference of the upper and lower functions over the interval where they intersect
• To find the area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$, where $f(x) \geq g(x)$ on $[a,b]$, use the formula $\int_a^b [f(x) - g(x)] dx$
• If the curves intersect at more than two points, the interval must be split into subintervals where one function is consistently above the other
• The area between two curves can also be found using horizontal rectangles, where the width is determined by the difference in $y$-values and the height is determined by the $x$-values
  • In this case, the roles of $x$ and $y$ are reversed, and the integration is performed with respect to $y$
• When finding the area between curves, it is essential to identify the points of intersection and determine which function is above the other in each subinterval
• Applications of area between curves include finding the area of irregular shapes, the region between intersecting graphs, and the area enclosed by polar curves

Volume of Solids

• The volume of a solid can be found by integrating the cross-sectional area of the solid over an interval
• The disk method calculates the volume of a solid of revolution formed by rotating a region bounded by $y=f(x)$, $y=0$, $x=a$, and $x=b$ about the $x$-axis using the formula $V = \int_a^b \pi [f(x)]^2 dx$
  • The disk method uses circular cross-sections perpendicular to the axis of rotation
• The washer method calculates the volume of a solid of revolution formed by rotating a region bounded by $y=f(x)$, $y=g(x)$, $x=a$, and $x=b$ about the $x$-axis using the formula $V = \int_a^b \pi ([f(x)]^2 - [g(x)]^2) dx$
  • The washer method uses washer-shaped cross-sections perpendicular to the axis of rotation
• The shell method calculates the volume of a solid of revolution by using cylindrical shells parallel to the axis of rotation
  • For a solid formed by rotating a region bounded by $x=f(y)$, $x=g(y)$, $y=c$, and $y=d$ about the $y$-axis, the volume is given by $V = \int_c^d 2\pi x[f(y) - g(y)] dy$
• The cross-section method calculates the volume of a solid by integrating the area of cross-sections perpendicular to a specified axis
  • The area of the cross-sections is typically a function of the position along the axis
• Applications of volume of solids include finding the volume of objects formed by rotating curves, the volume of irregular shapes, and the volume of solids with known cross-sections

Arc Length and Surface Area

• Arc length is the distance along a curve between two points
• To find the arc length of a curve $y=f(x)$ from $x=a$ to $x=b$, use the formula $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$
  • This formula is derived using the Pythagorean theorem and the concept of infinitesimal arc lengths
• For parametric curves given by $x=f(t)$ and $y=g(t)$, where $a \leq t \leq b$, the arc length is given by $L = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2} dt$
• The surface area of a solid of revolution can be found by integrating the surface area of infinitesimal strips formed by rotating the arc length about an axis
• For a curve $y=f(x)$ rotated about the $x$-axis from $x=a$ to $x=b$, the surface area is given by $SA = \int_a^b 2\pi f(x) \sqrt{1 + [f'(x)]^2} dx$
• For a curve $x=f(y)$ rotated about the $y$-axis from $y=c$ to $y=d$, the surface area is given by $SA = \int_c^d 2\pi f(y) \sqrt{1 + [f'(y)]^2} dy$
• Applications of arc length and surface area include finding the length of curves, the perimeter of irregular shapes, and the surface area of objects formed by rotating curves

Work and Fluid Pressure

• Work is the product of force and displacement in the direction of the force
• When a variable force $F(x)$ acts on an object moving along a straight line from $x=a$ to $x=b$, the work done is given by $W = \int_a^b F(x) dx$
• Hooke's Law states that the force exerted by a spring is proportional to its displacement from equilibrium, $F(x) = kx$, where $k$ is the spring constant
  • The work done in stretching or compressing a spring from $x=a$ to $x=b$ is given by $W = \int_a^b kx dx = \frac{1}{2}k(b^2 - a^2)$
• Fluid pressure is the force per unit area exerted by a fluid on a surface
• The hydrostatic pressure at a depth $h$ in a fluid with density $\rho$ is given by $P = \rho gh$, where $g$ is the acceleration due to gravity
• The force exerted by fluid pressure on a vertical surface is given by $F = \int_a^b P(y) w(y) dy$, where $P(y)$ is the pressure at depth $y$ and $w(y)$ is the width of the surface at depth $y$
• Applications of work and fluid pressure include calculating the work done by variable forces, the energy stored in springs, the force exerted by fluids on surfaces, and the design of dams and tanks

Center of Mass and Moments

• The center of mass is the point at which an object's mass can be considered to be concentrated
• For a thin rod of length $L$ with linear density $\rho(x)$, the center of mass is given by $\bar{x} = \frac{\int_0^L x\rho(x) dx}{\int_0^L \rho(x) dx}$
• For a planar lamina bounded by $y=f(x)$, $y=g(x)$, $x=a$, and $x=b$, with density $\rho(x,y)$, the center of mass is given by:
  • $\bar{x} = \frac{\int_a^b \int_{g(x)}^{f(x)} x\rho(x,y) dy dx}{\int_a^b \int_{g(x)}^{f(x)} \rho(x,y) dy dx}$
  • $\bar{y} = \frac{\int_a^b \int_{g(x)}^{f(x)} y\rho(x,y) dy dx}{\int_a^b \int_{g(x)}^{f(x)} \rho(x,y) dy dx}$
• Moments are the product of a force and its distance from a reference point or axis
• The first moment of a thin rod about the origin is given by $M = \int_a^b x\rho(x) dx$
• The second moment (moment of inertia) of a thin rod about the origin is given by $I = \int_a^b x^2\rho(x) dx$
• Applications of center of mass and moments include finding the balance point of objects, the stability of structures, and the rotational dynamics of rigid bodies

Real-World Applications

• Optimization problems involve finding the maximum or minimum value of a quantity subject to given constraints
  • Examples include maximizing profit, minimizing cost, or optimizing the dimensions of a container
• Growth and decay problems model situations where a quantity increases or decreases exponentially over time
  • The exponential growth model is given by $A(t) = A_0e^{kt}$, where $A_0$ is the initial amount and $k$ is the growth rate
  • The exponential decay model is given by $A(t) = A_0e^{-kt}$, where $A_0$ is the initial amount and $k$ is the decay rate
• Population dynamics models describe the change in a population over time, considering factors such as birth rates, death rates, and carrying capacity
  • The logistic growth model is given by $\frac{dP}{dt} = kP(1-\frac{P}{K})$, where $P$ is the population size, $k$ is the growth rate, and $K$ is the carrying capacity
• Carbon dating is a method for determining the age of organic materials based on the decay of radioactive carbon-14
  • The amount of carbon-14 remaining in a sample after time $t$ is given by $A(t) = A_0e^{-kt}$, where $A_0$ is the initial amount and $k$ is the decay constant
• Applications of integration in physics include finding the work done by a force, the potential energy of a system, and the electric potential and flux in electrostatics

Common Pitfalls and Tips

• When setting up integrals, ensure that the integrand and limits of integration are correctly identified based on the problem context
• Be cautious when using definite integrals to find net change, as the result may be positive or negative depending on the function and the order of the limits
• When finding the area between curves, correctly determine which function is above the other in each subinterval and use absolute values if necessary
• In volume problems, choose the appropriate method (disk, washer, shell, or cross-section) based on the geometry of the solid and the axis of rotation
• When using the shell method, ensure that the radius of the shell is correctly expressed in terms of the integration variable
• In arc length and surface area problems, verify that the integrand is non-negative to avoid extraneous solutions
• When solving work problems, ensure that the force function is expressed in terms of the displacement variable and that the limits of integration correspond to the starting and ending positions
• In center of mass and moment problems, correctly identify the density function and the region of integration based on the geometry of the object
• When applying integration to real-world problems, clearly define variables, state assumptions, and interpret the results in the context of the problem
• Practice a variety of problems to develop proficiency in recognizing patterns, selecting appropriate techniques, and applying the concepts of integration in different contexts