Course Cheatsheet

Section 06: Integral Calculus

Antiderivatives

Definition

Antiderivative: \(F(x)\) is an antiderivative of \(f(x)\) if \(F'(x) = f(x)\).

Indefinite Integral Notation: \[\int f(x) \, dx = F(x) + C\]

where \(C\) is the constant of integration.

Always Include +C

For indefinite integrals, always add the constant of integration!

Basic Antiderivative Rules

Function	Antiderivative
\(x^n\) (\(n \neq -1\))	\(\frac{x^{n+1}}{n+1} + C\)
\(\frac{1}{x}\)	\(\ln\|x\| + C\)
\(e^x\)	\(e^x + C\)
\(e^{ax}\)	\(\frac{1}{a}e^{ax} + C\)
\(a^x\)	\(\frac{a^x}{\ln a} + C\)
\(\sin x\)	\(-\cos x + C\)
\(\cos x\)	\(\sin x + C\)

Integration Rules

Rule	Formula
Constant Multiple	\(\int k \cdot f(x) \, dx = k \int f(x) \, dx\)
Sum/Difference	\(\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx\)

Memory Aid

“Integration undoes differentiation.” Ask: “What function, when differentiated, gives me this?”

Definite Integrals

Fundamental Theorem of Calculus

\[\int_a^b f(x) \, dx = F(b) - F(a)\]

where \(F(x)\) is any antiderivative of \(f(x)\).

Notation: \(\left[F(x)\right]_a^b = F(b) - F(a)\)

Evaluation Process

Always write out the bracket notation: \(\left[F(x)\right]_a^b = F(b) - F(a)\). This helps avoid sign errors when substituting limits.

Properties of Definite Integrals

Property	Formula
Constant Multiple	\(\int_a^b k \cdot f(x) \, dx = k \int_a^b f(x) \, dx\)
Sum	\(\int_a^b [f(x) + g(x)] \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx\)
Reversal	\(\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx\)
Additivity	\(\int_a^c f(x) \, dx = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx\)
Zero Width	\(\int_a^a f(x) \, dx = 0\)

Net vs. Total Area

Net area (signed): \(\int_a^b f(x) \, dx\) (areas below x-axis are negative)
Total area (unsigned): \(\int_a^b |f(x)| \, dx\)

Area Problems

Area Under a Curve

For \(f(x) \geq 0\) on \([a, b]\): \[\text{Area} = \int_a^b f(x) \, dx\]

For functions that cross the x-axis, split at zeros: \[\text{Total Area} = \int_a^c f(x) \, dx - \int_c^b f(x) \, dx\]

(where \(f(c) = 0\) and \(f(x) < 0\) on \((c, b)\))

Area Between Curves

For \(f(x) \geq g(x)\) on \([a, b]\): \[\text{Area} = \int_a^b [f(x) - g(x)] \, dx\]

Finding the Upper Function

Always determine which function is “on top” by testing a point in the interval: - If \(f(c) > g(c)\), then \(f\) is the upper function - Use: Upper minus Lower in the integrand

Steps for Area Between Curves

Find intersection points: Solve \(f(x) = g(x)\)
Determine which function is on top in each region
Set up integral(s): \(\int_a^b [\text{top} - \text{bottom}] \, dx\)
Evaluate using the Fundamental Theorem

Economic Applications

Consumer Surplus

\[CS = \int_0^{q^*} [D(q) - p^*] \, dq\]

where: - \(D(q)\) = demand function - \(q^*\) = equilibrium quantity - \(p^*\) = equilibrium price

Interpretation: Total benefit consumers receive from paying less than their maximum willingness to pay.

Geometric Interpretation

Consumer surplus is the area between the demand curve and the horizontal price line. Sketch the region to visualize what you’re calculating!

Producer Surplus

\[PS = \int_0^{q^*} [p^* - S(q)] \, dq\]

where: - \(S(q)\) = supply function - \(q^*\) = equilibrium quantity - \(p^*\) = equilibrium price

Interpretation: Total benefit producers receive from selling at a price higher than their minimum acceptable price.

Finding Equilibrium

At equilibrium: \(D(q) = S(q)\)

Solve to find \(q^*\), then \(p^* = D(q^*) = S(q^*)\)

Total Cost from Marginal Cost

\[C(x) = \int MC(x) \, dx = \int C'(x) \, dx\]

Use initial condition \(C(0) = \text{Fixed Costs}\) to find \(C\).

Revenue from Marginal Revenue

\[R(x) = \int MR(x) \, dx = \int R'(x) \, dx\]

Typically \(R(0) = 0\) (no revenue with no sales).

Average Value of a Function

\[f_{avg} = \frac{1}{b-a} \int_a^b f(x) \, dx\]

Interpretation: The constant height of a rectangle with the same area as the region under the curve.

Visual Understanding

The average value is like “flattening” the curve into a horizontal line with equal area.

Integration by Parts

\[\int u \, dv = uv - \int v \, du\]

LIATE Rule for Choosing u

Choose \(u\) from this list (first available):

Logarithmic functions (\(\ln x\))
Inverse trigonometric functions
Algebraic functions (\(x^n\), polynomials)
Trigonometric functions
Exponential functions (\(e^x\))

Common Integration by Parts Formulas

Integral	Result
\(\int x e^x \, dx\)	\(e^x(x - 1) + C\)
\(\int x^n \ln x \, dx\)	\(\frac{x^{n+1}}{n+1}\left(\ln x - \frac{1}{n+1}\right) + C\)
\(\int \ln x \, dx\)	\(x(\ln x - 1) + C\)

Initial Value Problems

Given: \(f'(x) = g(x)\) and \(f(a) = b\)

Steps:

Find general antiderivative: \(f(x) = \int g(x) \, dx = G(x) + C\)
Apply initial condition: \(f(a) = G(a) + C = b\)
Solve for \(C\)
Write specific solution

Quick Reference: Common Integrals

Function	Antiderivative
\(k\) (constant)	\(kx + C\)
\(x^n\)	\(\frac{x^{n+1}}{n+1} + C\) (\(n \neq -1\))
\(\frac{1}{x}\)	\(\ln\|x\| + C\)
\(e^x\)	\(e^x + C\)
\(e^{ax}\)	\(\frac{e^{ax}}{a} + C\)
\(\sqrt{x}\)	\(\frac{2x^{3/2}}{3} + C\)
\(\frac{1}{x^2}\)	\(-\frac{1}{x} + C\)

Problem-Solving Strategies

For Indefinite Integrals

Simplify first if possible (expand, factor out constants)
Rewrite radicals and fractions as powers
Apply rules term by term
Verify by differentiating the result

For Definite Integrals

Find the antiderivative (without +C)
Evaluate at upper limit
Evaluate at lower limit
Subtract: \(F(b) - F(a)\)

For Area Problems

Sketch the region (if possible)
Find boundaries and intersection points
Identify upper and lower functions
Set up the integral(s)
Calculate and interpret

Common Mistakes to Avoid

Forgetting the \(+C\) for indefinite integrals
Wrong sign when function is below x-axis
Mixing up upper and lower functions in area problems
Forgetting to apply both limits in definite integrals
Not simplifying before integrating