Integration Techniques and Applications in Business Calculus (Chapters 7.1–8.4)

Integration Techniques and Applications

Definite and Indefinite Integrals

Integration is a fundamental concept in calculus, used to find areas, accumulated quantities, and solve various applied problems. Integrals can be classified as definite (with limits of integration) or indefinite (without limits, representing a family of functions).

• Basic Integration Rules (7.1): Use standard formulas for polynomials, exponentials, and trigonometric functions.

• Substitution (7.2): Used when the integrand contains a function and its derivative. Set u = g(x) to simplify the integral.

• Integration by Parts (8.1): Used for products of functions, based on the formula: $\int u \, dv = uv - \int v \, du$

• Definite Integrals (7.4): Evaluate the integral between two bounds a and b: $\int_a^b f(x) \, dx$

Example: Find $\int x^2 \, dx = \frac{1}{3}x^3 + C$

Riemann Sums (7.3)

Riemann sums approximate the area under a curve by summing the areas of rectangles. The choice of endpoints (left or right) affects the approximation.

• Delta x ($\Delta x$): The width of each subinterval: $\Delta x = \frac{b-a}{n}$

• Left Endpoint: Use the left value of each subinterval for the rectangle's height.

• Right Endpoint: Use the right value of each subinterval for the rectangle's height.

Example: Approximate $\int_0^2 x^2 \, dx$ using 4 rectangles and left endpoints.

Area Under Curves and Between Curves (7.5)

The area under a curve or between two curves can be found using definite integrals.

• Area under a curve: $\int_a^b f(x) \, dx$

• Area between curves: $\int_a^b [f(x) - g(x)] \, dx$, where $f(x)$ is above $g(x)$ on [a, b]

• If part of the graph is below the x-axis, the integral gives a negative value; take the absolute value for area.

• If limits a and b are not given, solve $f(x) = g(x)$ to find intersection points.

Example: Find the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$:

$\int_0^1 (x - x^2) \, dx$

Applications of Integrals

• Consumers’ Surplus (7.5): Measures the benefit to consumers; formula: $\int_0^{q_0} [D(q) - p_0] \, dq$

• Producers’ Surplus (7.5): Measures the benefit to producers; formula: $\int_0^{q_0} [p_0 - S(q)] \, dq$

• Average Value (8.2): The average value of $f(x)$ on $[a, b]$: $\frac{1}{b-a} \int_a^b f(x) \, dx$

• Present Value of Money Flow (8.3): The present value of a continuous income stream: $\int_0^T f(t) e^{-rt} \, dt$

• Accumulated Amount of Money Flow (8.3): The future value of a continuous income stream: $e^{rT} \int_0^T f(t) e^{-rt} \, dt$

Example: If $f(t)$ is the rate of income at time $t$, $r$ is the interest rate, and $T$ is the time period, the present value is calculated as above.

Improper Integrals (8.4)

Improper integrals involve infinite limits or unbounded integrands. They are evaluated as limits.

• Convergence: The integral converges if the limit exists and is finite.

• Divergence: The integral diverges if the limit does not exist or is infinite.

Example: $\int_1^{\infty} \frac{1}{x^2} \, dx$ converges, but $\int_1^{\infty} \frac{1}{x} \, dx$ diverges.

Choosing an Integration Technique

To solve an integral, first check if algebraic simplification is possible. Then, choose the appropriate technique:

• Basic Rules: Use for simple polynomials, exponentials, etc.

• Substitution: Use when the integrand contains a function and its derivative.

• Integration by Parts: Use for products of functions (e.g., $x e^x$, $x \sin x$).

Tip: Look for patterns such as a function multiplied by its derivative (substitution), or a product of algebraic and transcendental functions (integration by parts).

Key Formulas and Variables

The following table summarizes the main formulas and their variables:

Formula	Application	Variables
$\int_0^{q_0} [D(q) - p_0] \, dq$	Consumers' Surplus	$D(q)$: demand function, $p_0$: equilibrium price, $q_0$: equilibrium quantity
$\int_0^{q_0} [p_0 - S(q)] \, dq$	Producers' Surplus	$S(q)$: supply function, $p_0$: equilibrium price, $q_0$: equilibrium quantity
$\frac{1}{b-a} \int_a^b f(x) \, dx$	Average Value	$f(x)$: function, $a$, $b$: interval endpoints
$\int_0^T f(t) e^{-rt} \, dt$	Present Value	$f(t)$: income rate, $r$: interest rate, $T$: time period
$e^{rT} \int_0^T f(t) e^{-rt} \, dt$	Accumulated Amount	$f(t)$: income rate, $r$: interest rate, $T$: time period

Sample Solutions to Selected Problems

• Indefinite Integral: $\int (8-x)^{3/2} \, dx = -\frac{2x}{5}(8-x)^{5/2} - \frac{4}{35}(8-x)^{7/2} + C$

• Logarithmic Integral: $\int \frac{1}{25-9x^2} \, dx = -\frac{1}{18} \ln|25-9x^2| + C$

• Exponential Integral: $\int x e^x \, dx = x e^x - e^x + C$

• Improper Integral: If the integral diverges, state "diverges" (e.g., $\int_1^{\infty} \frac{1}{x} \, dx$ diverges).

• Definite Integral Application: If the answer is a value (e.g., 5000 gallons), interpret in context (e.g., total accumulated quantity).

Additional Study Recommendations

• Practice identifying which integration technique to use for a given integral.

• Review textbook exercises and solutions, especially those involving mixed techniques.

• Understand the meaning and application of each formula, not just how to compute it.

Additional info: This guide expands on the review outline by providing definitions, formulas, and examples for each topic. For more practice, refer to the suggested textbook problems and answer keys.