∫Calculus I Unit 6 – Applications of Integration

Key Concepts and Definitions

• Integration involves finding the area under a curve, which represents the accumulation of a quantity over an interval
• Definite integrals calculate the area between a function and the x-axis over a specific interval $[a, b]$, denoted as $\int_a^b f(x) dx$
• Indefinite integrals find the set of all antiderivatives of a function, denoted as $\int f(x) dx = F(x) + C$, where $C$ is an arbitrary constant
  • Antiderivatives are functions whose derivative is the original function
• The Fundamental Theorem of Calculus connects differentiation and integration, establishing their inverse relationship
• Riemann sums approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles
  • As the number of subintervals approaches infinity, the Riemann sum converges to the definite integral
• Integration techniques include substitution, integration by parts, trigonometric substitution, and partial fractions
• Applications of integration include finding areas between curves, volumes of solids, arc lengths, surface areas, work, fluid pressure, center of mass, and moments

Fundamental Theorem of Calculus Review

• 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 $f$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$
  • This theorem allows us to evaluate definite integrals using antiderivatives
• The Second Fundamental Theorem of Calculus states that if $f$ is continuous on an open interval $I$ containing $a$, then the function $g(x) = \int_a^x f(t) dt$ is continuous on $I$ and differentiable on the interior of $I$, with $g'(x) = f(x)$
  • This theorem relates the definite integral to the derivative of a function
• The FTC enables us to find the exact value of a definite integral without using Riemann sums or approximations
• 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 crucial for solving various applications of integration, such as finding areas, volumes, and lengths

Area Between Curves

• Finding the area between curves involves integrating the difference between two functions over a given interval
• To find the area between two curves $y = f(x)$ and $y = g(x)$ over the interval $[a, b]$, integrate the absolute value of their difference: $A = \int_a^b |f(x) - g(x)| dx$
  • If $f(x) \geq g(x)$ on $[a, b]$, then $A = \int_a^b [f(x) - g(x)] dx$
  • If $f(x) \leq g(x)$ on $[a, b]$, then $A = \int_a^b [g(x) - f(x)] dx$
• When the curves intersect, split the interval at the intersection points and calculate the area of each subregion separately
• To find the area between two curves given by $x = f(y)$ and $x = g(y)$ over the interval $[c, d]$, integrate the absolute value of their difference with respect to $y$: $A = \int_c^d |f(y) - g(y)| dy$
• Be cautious of the order of the functions when setting up the integral to ensure the area is positive
• Sketch the curves to visualize the region and determine the appropriate integration limits and function order

Volume of Solids

• Integration can be used to calculate the volume of solids formed by revolving a region bounded by curves around an axis
• The disk method calculates the volume of a solid formed by revolving a region around the x-axis or y-axis
  • For a region bounded by $y = f(x)$, $y = g(x)$, $x = a$, and $x = b$, the volume of the solid formed by revolving the region around the x-axis is $V = \pi \int_a^b [f(x)]^2 - [g(x)]^2 dx$
  • For a region bounded by $x = f(y)$, $x = g(y)$, $y = c$, and $y = d$, the volume of the solid formed by revolving the region around the y-axis is $V = \pi \int_c^d [f(y)]^2 - [g(y)]^2 dy$
• The washer method calculates the volume of a solid formed by revolving a region around a line parallel to the x-axis or y-axis
  • For a region bounded by $y = f(x)$, $y = g(x)$, $x = a$, and $x = b$, the volume of the solid formed by revolving the region around the line $y = k$ is $V = \pi \int_a^b [f(x) - k]^2 - [g(x) - k]^2 dx$
  • For a region bounded by $x = f(y)$, $x = g(y)$, $y = c$, and $y = d$, the volume of the solid formed by revolving the region around the line $x = h$ is $V = \pi \int_c^d [f(y) - h]^2 - [g(y) - h]^2 dy$
• The shell method calculates the volume of a solid formed by revolving a region around the y-axis or x-axis using cylindrical shells
  • For a region bounded by $y = f(x)$, $y = g(x)$, $x = a$, and $x = b$, the volume of the solid formed by revolving the region around the y-axis using the shell method is $V = 2\pi \int_a^b x[f(x) - g(x)] dx$
  • For a region bounded by $x = f(y)$, $x = g(y)$, $y = c$, and $y = d$, the volume of the solid formed by revolving the region around the x-axis using the shell method is $V = 2\pi \int_c^d y[f(y) - g(y)] dy$

Arc Length and Surface Area

• Arc length is the distance along a curve between two points
• To find the arc length of a curve $y = f(x)$ over the interval $[a, b]$, 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 a curve as a limit of line segments
• For a curve given parametrically by $x = f(t)$ and $y = g(t)$, where $a \leq t \leq b$, the arc length is $L = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2} dt$
• Surface area is the area of the surface formed by revolving a curve around an axis
• To find the surface area of a surface formed by revolving a curve $y = f(x)$ around the x-axis over the interval $[a, b]$, use the formula $SA = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} dx$
  • This formula is derived by considering the surface as a sum of the lateral areas of frustums of cones
• For a surface formed by revolving a curve given parametrically by $x = f(t)$ and $y = g(t)$, where $a \leq t \leq b$, around the x-axis, the surface area is $SA = 2\pi \int_a^b g(t) \sqrt{[f'(t)]^2 + [g'(t)]^2} dt$
• Similar formulas exist for surfaces formed by revolving curves around the y-axis or other lines parallel to the axes

Work and Fluid Pressure

• Work is the product of force and displacement in the direction of the force
• To find the work done by a variable force $F(x)$ acting on an object moving along a straight line from $x = a$ to $x = b$, use the formula $W = \int_a^b F(x) dx$
  • This formula is derived by considering work as the limit of the sum of the products of force and displacement over small intervals
• Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from equilibrium, $F(x) = kx$, where $k$ is the spring constant
  • The work done by a spring force in stretching or compressing a spring from $x = a$ to $x = b$ is $W = \int_a^b kx dx = \frac{1}{2}k(b^2 - a^2)$
• Fluid pressure is the force per unit area exerted by a fluid on a surface
• To find the force exerted by a fluid with density $\rho$ on a vertical rectangular plate with width $w$ and height $h$, submerged so that its top edge is at depth $a$ and its bottom edge is at depth $b$, use the formula $F = \rho g w \int_a^b x dx = \frac{1}{2}\rho g w (b^2 - a^2)$, where $g$ is the acceleration due to gravity
  • This formula is derived by integrating the pressure, which varies linearly with depth, over the surface area of the plate

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 certain calculations
• For a thin rod of length $L$ with linear density $\rho(x)$, the x-coordinate of the center of mass is given by $\bar{x} = \frac{\int_0^L x\rho(x) dx}{\int_0^L \rho(x) dx}$
  • This formula is derived by considering the rod as a sum of infinitesimal mass elements and finding the balance point
• For a thin plate in the xy-plane with surface density $\sigma(x, y)$, the coordinates of the center of mass are given by $\bar{x} = \frac{\iint_R x\sigma(x, y) dA}{\iint_R \sigma(x, y) dA}$ and $\bar{y} = \frac{\iint_R y\sigma(x, y) dA}{\iint_R \sigma(x, y) dA}$, where $R$ is the region occupied by the plate
• Moments are quantities that describe the distribution of mass or force about a point or axis
• The moment of a force $F$ about a point is the product of the force and the perpendicular distance from the point to the line of action of the force
• The first moment of area of a region $R$ in the xy-plane about the x-axis is given by $Q_x = \iint_R y dA$, and the first moment of area about the y-axis is given by $Q_y = \iint_R x dA$
  • These moments are used in calculating centroids and in beam bending problems

Real-World Applications

• Optimization problems involve finding the maximum or minimum value of a quantity subject to certain constraints
  • Example: Minimizing the surface area of a cylindrical can with a fixed volume to minimize the cost of materials
• Hydrostatic force and pressure calculations are used in designing dams, tanks, and other structures that hold fluids
  • Example: Calculating the force on a submerged gate of a dam to determine the required strength of the gate
• Center of mass calculations are important in designing balanced structures and understanding the stability of objects
  • Example: Finding the center of mass of an irregularly shaped object to determine its balance point
• Moments of inertia, which involve integration, are used in analyzing the rotational motion and bending of objects
  • Example: Calculating the moment of inertia of a beam to determine its resistance to bending under a load
• Physics applications include work, energy, and power calculations involving forces and displacements
  • Example: Determining the work done by a variable force in moving an object along a path
• Biology and medicine applications include calculating the cardiac output of the heart and the absorption of drugs in the bloodstream
  • Example: Modeling the concentration of a drug in the bloodstream over time using integration
• Economics applications include calculating consumer and producer surplus, marginal revenue, and marginal cost
  • Example: Finding the consumer surplus, which is the area between the demand curve and the price line