➗Calculus II Unit 2 – Applications of Integration

Key Concepts and Definitions

• Integration involves finding the area under a curve, the opposite of differentiation which finds the slope of a tangent line at a point
• Definite integrals calculate the area under a curve between two specific points (lower and upper limits of integration)
• Indefinite integrals find the general antiderivative of a function without specific limits
  • Represented as $\int f(x) dx = F(x) + C$, where $C$ is the constant of integration
• Riemann sums approximate the area under a curve by dividing it into rectangles and summing their areas
  • As the number of rectangles approaches infinity, the Riemann sum approaches the exact value of the definite integral
• Integration by substitution is a technique for evaluating integrals by substituting a new variable to simplify the expression
• Integration by parts is a method for integrating products of functions, using the formula $\int u dv = uv - \int v du$
• Improper integrals are integrals with infinite limits or discontinuous integrands
  • Convergent improper integrals have finite values, while divergent improper integrals do not

Fundamental Theorem of Calculus Review

• The Fundamental Theorem of Calculus (FTC) connects differentiation and integration, showing they are inverse operations
• The First Fundamental Theorem of Calculus states that if $f$ is continuous on $[a, b]$, then the function $g(x) = \int_a^x f(t) dt$ is an antiderivative of $f$ on $[a, b]$
  • In other words, $\frac{d}{dx} \int_a^x f(t) dt = f(x)$
• The Second Fundamental Theorem of Calculus provides a way to evaluate definite integrals using antiderivatives
  • If $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$
• The FTC allows us to calculate definite integrals without using Riemann sums or other approximation methods
• The Mean Value Theorem for Integrals is a consequence of the FTC, stating that for a continuous function $f$ on $[a, b]$, there exists a point $c \in [a, b]$ such that $\int_a^b f(x) dx = f(c)(b - a)$

Area Between Curves

• To find the area between two curves, integrate the difference of their functions over the interval where they intersect
  • If $f(x) \geq g(x)$ on $[a, b]$, then the area between the curves is $\int_a^b [f(x) - g(x)] dx$
• When the curves intersect at more than two points, split the integral into subintervals where one function is consistently greater than the other
• For curves defined parametrically, find the intersection points by setting their x and y components equal and solving for the parameter
• When working with polar curves, use the formula $A = \frac{1}{2} \int_a^b [r_2(\theta)^2 - r_1(\theta)^2] d\theta$ to find the area between them
• Be careful with the order of subtraction when setting up the integral, as subtracting in the wrong order can lead to a negative area

Volume of Solids

• The disk method calculates the volume of a solid of revolution by integrating the area of circular cross-sections perpendicular to the axis of rotation
  • For a solid formed by rotating the region bounded by $y = f(x)$, $y = 0$, $x = a$, and $x = b$ about the x-axis, the volume is $V = \int_a^b \pi [f(x)]^2 dx$
• The washer method is similar to the disk method but subtracts the volume of an inner hole from each cross-section
  • For a solid formed by rotating the region bounded by $y = f(x)$ and $y = g(x)$ about the x-axis, the volume is $V = \int_a^b \pi ([f(x)]^2 - [g(x)]^2) dx$
• The shell method calculates volume by integrating cylindrical shells parallel to the axis of rotation
  • For a solid formed by rotating the region bounded by $x = f(y)$, $x = g(y)$, $y = c$, and $y = d$ about the y-axis, the volume is $V = 2\pi \int_c^d x[f(y) - g(y)] dy$
• When using the shell method, the radius of each shell is the distance from the shell to the axis of rotation
• Choose the most appropriate method based on the geometry of the solid and the axis of rotation to simplify the integral

Arc Length and Surface Area

• Arc length measures the distance along a curve between two points
  • For a function $y = f(x)$ on $[a, b]$, the arc length is $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$
  • For a parametric curve $(x(t), y(t))$ on $[a, b]$, the arc length is $L = \int_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2} dt$
• Surface area is the area of the surface formed by rotating a curve about an axis
  • For a curve $y = f(x)$ rotated about the x-axis on $[a, b]$, the surface area is $SA = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} dx$
  • For a parametric curve $(x(t), y(t))$ rotated about the x-axis on $[a, b]$, the surface area is $SA = 2\pi \int_a^b y(t) \sqrt{[x'(t)]^2 + [y'(t)]^2} dt$
• When calculating surface area, the integrand represents the circumference of each circular strip multiplied by the arc length of the generating curve
• Be careful to use the correct formula based on whether the curve is given as an explicit function or parametrically

Work and Fluid Pressure

• Work is the product of force and displacement in the direction of the force
  • For a variable force $F(x)$ acting on an object moving along a straight line from $x = a$ to $x = b$, the work done is $W = \int_a^b F(x) dx$
• 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 to compress or stretch a spring from $a$ to $b$ 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
  • The hydrostatic pressure at a depth $h$ in a fluid with density $\rho$ is $P = \rho gh$, where $g$ is the acceleration due to gravity
• The force exerted by fluid pressure on a vertical surface is $F = \int_a^b P(y) w(y) dy$, where $P(y)$ is the pressure at depth $y$ and $w(y)$ is the width of the surface at that depth
• Work and fluid pressure problems often involve setting up integrals based on physical principles and given information

Center of Mass and Moments

• The center of mass is the point where an object's mass is evenly distributed, and the object would balance if supported at that point
  • For a thin rod with linear density $\rho(x)$ along the x-axis from $a$ to $b$, the x-coordinate of the center of mass is $\bar{x} = \frac{\int_a^b x\rho(x) dx}{\int_a^b \rho(x) dx}$
  • For a lamina (thin plate) with surface density $\sigma(x, y)$ over a region $R$, the coordinates of the center of mass are $\bar{x} = \frac{\iint_R x\sigma(x, y) dA}{\iint_R \sigma(x, y) dA}$ and $\bar{y} = \frac{\iint_R y\sigma(x, y) dA}{\iint_R \sigma(x, y) dA}$
• Moments measure the tendency of a force to cause rotation about a point or axis
  • The moment of a point mass $m$ about the origin is $\vec{r} \times m\vec{v}$, where $\vec{r}$ is the position vector and $\vec{v}$ is the velocity vector
  • For a lamina with surface density $\sigma(x, y)$, the moments of inertia about the x and y axes are $I_x = \iint_R y^2 \sigma(x, y) dA$ and $I_y = \iint_R x^2 \sigma(x, y) dA$
• The parallel axis theorem relates the moment of inertia about an axis passing through the center of mass to the moment of inertia about a parallel axis
  • If $I_{cm}$ is the moment of inertia about an axis through the center of mass and $d$ is the perpendicular distance between the axes, then the moment of inertia about the parallel axis is $I = I_{cm} + md^2$

Real-World Applications and Examples

• Calculating the volume of irregular objects (vases, bottles, or other containers) by rotating their cross-sections
• Determining the work done by variable forces (spring compression, cable hoisting, or fluid displacement)
• Finding the center of mass and moments of inertia for complex shapes (beams, plates, or mechanical components)
• Modeling population growth or decay using exponential and logistic functions
• Analyzing the flow of fluids through pipes or the pressure exerted on dams and tanks
• Calculating the length of power cables or pipelines spanning uneven terrain
• Determining the surface area of objects for coating, heating, or cooling applications
• Estimating the mass of thin wires, rods, or sheets with non-uniform density
• Optimizing the design of containers, beams, or other structures to minimize weight or maximize strength
• Predicting the motion of objects under the influence of variable forces or moments