Integration and Its Applications: Area and the Fundamental Theorem of Calculus

Area and Definite Integrals

• Geometric vs. Calculus-Based Area: In geometry, area is defined as a number representing the size of a bounded region. Standard regions (rectangles, triangles, circles) have established formulas. Calculus allows for the calculation of the area of nonstandard regions, such as the region $R$ bounded by a curve and the $x$-axis.

• Definition of a Definite Integral: Let $f$ be a nonnegative and continuous function on the closed interval $[a, b]$. The area of the region bounded by the graph of $f$, the $x$-axis, and the vertical lines $x = a$ and $x = b$ is denoted by the definite integral:     Area = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } \text{Area} = \text{Area} = \beta \text{ is denoted by } 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ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta.\n\n* **Notation**: The term eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta$is the definite integral from$a$to$b. \n * a: Lower limit of integration.\n * b: Upper limit of integration.\n\n* **Example 1: Geometric Evaluation**: Evaluate the definite integral representing the area bounded by f(x) = 2x$, the$x$-axis, and$x = 2.\n * The region is triangular.\n * Height: f(2) = 2(2) = 4 ext{ units}.\n * Base: 2 ext{ units}$(from$x=0$to$x=2).\n * ext{Area} = rac{1}{2} imes ext{base} imes ext{height} = rac{1}{2}(2)(4) = 4.\n\n# The Fundamental Theorem of Calculus (FTC)\n\n* **Discovery of the Relationship**: To understand the connection between area and calculus, consider a function A(x)$representing the area from a constant$a$to a variable$x.\n * If x$increases by$ riangle x$, the area increases by$ riangle A.\n * Let f(m)$and$f(M)$be the minimum and maximum values of$f$on the interval$[x, x + riangle x].\n * The change in area is bounded: f(m) riangle x riangle A riangle f(M) riangle x.\n * Dividing by riangle x$gives:$f(m) riangle rac{ riangle A}{ riangle x} riangle f(M).\n * As riangle x ightarrow 0$, both$f(m)$and$f(M)$approach$f(x)$. Therefore,$ rac{dA}{dx} = f(x).\n * This implies that A(x)$is an antiderivative of$f(x).\n\n* **The Theorem Defined**: If f$is continuous on a closed interval$[a, b]$, then the definite integral of$f$from$a$to$b is:\n    eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta = F(b) - F(a)\n    where F$is any antiderivative of$f.\n\n* **Guidelines for Application**:\n 1. **Purpose**: The FTC is a method for evaluation, not a procedural instruction for finding antiderivatives themselves.\n 2. **Notation**: Use the vertical bar notation: eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta = [F(x)]a^b = F(b) - F(a).\n 3. **Dropping the Constant (C$)**: The constant of integration$C$is omitted in definite integrals because it cancels out during subtraction:$(F(b) + C) - (F(a) + C) = F(b) - F(a).\n\n* **Note on Definite Integral Values**: Unlike geometric area (which is always positive), a definite integral can be positive, negative, or zero.\n\n# Properties of Definite Integrals\n\nAssuming f$and$g$are continuous on$[a, b]:\n\n1. **Constant Multiple Rule**: \n    eta ext{ is denoted by } ext{Area} = ext{Area} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta k f(x) dx = k eta ext{ is denoted by } ext{Area} = f(x) dx$(where$k is a constant).\n2. **Sum and Difference Rule**:\n    eta ext{ is denoted by } ext{Area} = ext{Area} = eta [f(x) riangle g(x)] dx = eta ext{ is denoted by } ext{Area} = f(x) dx riangle eta ext{ is denoted by } ext{Area} = g(x) dx.\n3. **Additive Property of Intervals**:\n    eta ext{ is denoted by } ext{Area} = ext{Area} = eta f(x) dx = eta ext{ is denoted by } ext{Area} = ext{Area} = eta f(x) dx + eta ext{ is denoted by } ext{Area} = ext{Area} = eta f(x) dx$for$a < c < b.\n4. **Zero Interval**: \n    eta ext{ is denoted by } ext{Area} = ext{Area} = eta f(x) dx = 0.\n5. **Reversal of Limits**: \n    eta ext{ is denoted by } ext{Area} = ext{Area} = eta f(x) dx = - eta ext{ is denoted by } ext{Area} = ext{Area} = eta f(x) dx.\n\n# Marginal Analysis\n\n* **Concept**: Integration is the reverse process of finding marginal values. Definite integrals are used to find the exact increase or decrease in cost (C$), revenue ($R$), or profit ($P) resulting from producing/selling additional units.\n\n* **Example 6: Change in Profit**: Marginal profit is modeled by P'(x) = -0.0005x + 12.2.\n * **Part A**: Sales increase from 100$to$101 units.\n         ext{Change in Profit} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta (-0.0005x + 12.2) dx = [-0.00025x^2 + 12.2x]{100}^{101} riangle ext{\$}12.15.\n * **Part B**: Sales increase from 100$to$110 units.\n         ext{Change in Profit} = eta ext{ is denoted by } ext{Area} = ext{Area} = eta (-0.0005x + 12.2) dx = [-0.00025x^2 + 12.2x]_{100}^{110} riangle ext{\$}121.48.\n\n# Average Value of a Function\n\n* **Definition**: If f$is continuous on the interval$[a, b], the average value is given by:\n     ext{Average Value} = rac{1}{b - a} eta ext{ is denoted by } ext{Area} = f(x) dx.\n\n* **Example 7: World Population**: World population P$(in billions) from 1951 through 2000 is modeled ($t=1 is 1951). Find the average population from 1951 through 1975.\n * Interval: [1, 25].\n * Calculation: Integrate P$over$[1, 25]$and multiply by$ rac{1}{25 - 1} = rac{1}{24}.\n * Result: The average population was approximately 3.3 ext{ billion people}.\n\n# Even and Odd Functions\n\n* **Even Functions**: Symmetric with respect to the y$-axis. Condition:$f(-x) = f(x).\n * **Integration Rule**: If f$is even, then$eta ext{ is denoted by } ext{Area} = ext{Area} = eta f(x) dx = 2 eta ext{ is denoted by } ext{Area} = f(x) dx.\n\n* **Odd Functions**: Symmetric with respect to the origin. Condition: f(-x) = -f(x).\n * **Integration Rule**: If f$is odd, then$eta ext{ is denoted by } ext{Area} = ext{Area} = eta f(x) dx = 0.\n\n* **Example 8**:\n * eta ext{ is denoted by } ext{Area} = ext{Area} = eta x^2 dx$: Since$x^2$is even, this equals$2 eta ext{ is denoted by } ext{Area} = x^2 dx.\n * eta ext{ is denoted by } ext{Area} = ext{Area} = eta x^3 dx$: Since$x^3$is odd, the result is exactly$0.\n\n# Annuities\n\n* **Definitions**:\n * **Annuity**: A sequence of equal payments made at regular time intervals (e.g., payroll savings, home mortgage, IRAs).\n * **Amount of an Annuity**: The sum of all payments plus the interest earned.\n\n* **Annuity Formula (Continuous)**:\n    If c(t)$is the continuous income function (dollars per year),$r$is the decimal interest rate compounded continuously, and$T is the term in years:\n     ext{Amount} = e^{rT} eta ext{ is denoted by } ext{Area} = c(t)e^{-rt} dt.\n\n* **Example 9: IRA Deposit**: Deposit ext{\$}2000$annually for$15 ext{ years}$at$5\% interest.\n * Values: c(t) = 2000$,$r = 0.05$,$T = 15.\n * Set-up: ext{Amount} = e^{(0.05)(15)} eta ext{ is denoted by } ext{Area} = 2000e^{-0.05t} dt.\n * Simplified: 2000 e^{0.75} [ rac{e^{-0.05t}}{-0.05} ]_0^{15}.\n * Result: Approximately ext{\$}44,680.00$$.