Calculus - Integration

Mathematics II

Symmetric Functions (Odd and Even)

• Definition: A function is even if .

• Graphical Interpretation: The graph is symmetric about the y -axis.

• Integration Property: For

• Definition: A function is odd if .

• Graphical Interpretation: The graph is symmetric about the origin.

• Integration Property: For

1. Symmetry Check: . The function is even.

2. Apply Integration Property:

1. Integrate:

4. Evaluate:

Area Between Two Curves

Vertical Area Between Curves

• Formula: The area is given by:

• To Solve:

1. Find the points of intersection by solving . These points may define your limits of integration or indicate where the “top” and “bottom” functions switch.

2. Determine which function is greater ( or ) over each interval defined by the intersection points and limits.

3. Integrate the difference between the upper and lower function over each interval and sum the results.

1. Find Intersection Points: . Intersection points are and . These are also the given limits of integration.

2. Determine Upper/Lower Function: For , let’s test . Since in this interval, is the upper curve.

3. Set up and Calculate Integral:

Horizontal Area Between Curves

• Concept: If it’s easier to express the curves as and , you can integrate with respect to .

• Formula: The area between and from to is given by:

• To Solve: Similar to the vertical case, but solve for in terms of and integrate with respect to .

Integration by Substitution

• When to Use: An integral has the form .

• Steps:

1. Choose a substitution: Let .

2. Find the differential: Calculate .

3. Transform the integral: Substitute and into the integral. The integral should now be entirely in terms of .

4. Integrate with respect to : If .

5. Resubstitute: Replace with to get the final answer in terms of .

• For Definite Integrals:

1. Follow steps 1-3.

2. Change the limits of integration: If the original limits were and , the new limits will be and .

3. Integrate with respect to using the new limits: .

1. Substitution: Let .

2. Differential: . Rearrange to .

3. Transform: The integral becomes .

4. Integrate: .

5. Resubstitute: .

1. Substitution: Let .

2. Differential: . Rearrange to .

3. Change Limits: When , . When , .

4. Transform and Integrate:

5. Evaluate:

Riemann Sums and the Definite Integral

• Area Under a Curve: The area under from to can be approximated by summing the areas of rectangular strips of width . The height of each strip is , where is a point within the th subinterval.

• Riemann Sum:

• Definite Integral (by Definition): The exact area is found by taking the limit as the number of strips approaches infinity (and thus the width of each strip approaches zero):

• Approximation Types: ∘ Right-endpoint: ∘ Left-endpoint:

• Midpoint:

1. Identify parameters: .

2. Calculate .

3. Determine : Using right endpoints, .

4. Set up the Riemann Sum:

5. Evaluate the Sum:

6. Take the Limit:

Properties of Definite Integrals

• Equal Limits:

• Reversed Limits:

• Splitting an Integral: (for )

• Scalar Multiple:

• Addition/Subtraction:

• Linear Combination:

• Comparison Theorems:

• If on , then .

• If on , then .

• If on , then .

The Fundamental Theorem of Calculus (FTOC)

• FTOC Part 1 (Derivative of an Integral): If , then

• FTOC Part 2 (Evaluating Definite Integrals): If (i.e., is an antiderivative of ), then

1. Find an antiderivative: The antiderivative of is .

2. Apply FTOC:

Indefinite Integrals

• Definition: , where and is the constant of integration.

• Significance: It represents a family of functions whose derivatives are . The constant accounts for the fact that the derivative of a constant is zero.

Common Antiderivatives (derived from differentiation rules):

• (for )

Integration by Parts

• Formula (Standard):

• Formula (Leibniz Notation): Let and . Then and .

When to Use:

• When one part of the integrand ( or ) can be differentiated to zero after a few steps (e.g., polynomials).

• When one part ( or ) is easily integrable, and the other part ( or ) is differentiable.

• When the integral cannot be solved by substitution.

• Tabular Method: This is a systematic way to perform integration by parts when multiple steps are needed.

1. Create a table with columns for “Sign” (+/-), “f” (function to differentiate), “g” (function to integrate), “f’” (derivative of f), “sg” (integral of g), ” g” (second integral of g), etc.

2. Choose such that its derivatives eventually become zero.

3. The result is the sum of the products of the diagonal elements, with alternating signs, plus the integral of the product of the last row’s ” ” derivative and ” ” integral (if needed).

1. Choose Functions: Let (will differentiate to 0 ) and (easy to integrate). Then and .

2. Apply Integration by Parts (Tabular Method):

1. Combine Diagonals:

1. Choose Functions: We don’t have a product of two functions directly, but we can think of as . Let (differentiates nicely) and (easy to integrate). Then and .

2. Apply Integration by Parts:

Improper Integrals

1. At least one limit of integration is infinite.

2. The integrand has a vertical asymptote within the interval of integration.

Type 1: Infinite Limits of Integration

• Form: , or .

• Evaluation: Use limits.

• (split at any convenient point )

• Convergence/Divergence: If the limit exists and is finite, the integral converges. Otherwise, it diverges.

1. Set up with limit:

1. Integrate:

1. Evaluate Limit:

Type 2: Integrand with Asymptote

• Form: where has a vertical asymptote at some in .

• Evaluation: Split the integral at the point of discontinuity.

• If asymptote is at .

• Evaluate each part using limits:

• Convergence/Divergence: For the overall integral to converge, both split integrals must converge.

1. Identify Asymptote: The integrand has a vertical asymptote at .

2. Split and Use Limits:

1. Integrate:

1. Evaluate Limit:

Integration by Partial Fractions

• When to Use: Integrals of the form where can be factored into linear or irreducible quadratic factors.

• Steps:

1. Factor the Denominator: Factor completely into linear factors ( ) and irreducible quadratic factors .

2. Decompose: Set up the partial fraction decomposition:

• For each distinct linear factor , use a term .

• For each repeated linear factor , use terms .

• For each irreducible quadratic factor , use a term .

1. Solve for Coefficients: Multiply both sides of the decomposition by the original denominator to clear fractions. Then, solve for the unknown coefficients (A, B, etc.) by either:

• Strategic Substitution: Plugging in the roots of the factors.

• Equating Coefficients: Expanding both sides and matching coefficients of like powers of .

1. Integrate: Integrate the resulting sum of simpler fractions.

• Integrals of Decomposed Terms:

• (for )

• often requires completing the square and using substitution or trigonometric substitution.

1. Factor Denominator: .

2. Decompose:

1. Solve for Coefficients: Multiply by : . ∘ Let . Let .

2. Integrate:

Multivariate Functions

Domain Constraints

• Example: For , the domain requires . This is the region above or on the line .

• Example: For , the domain requires , which simplifies to . This is the interior of an ellipse.

Graphing 2-Variate Functions with Level Curves (Contour Maps)

• Concept: The graph of is a surface in . To visualize this, we can plot its “level curves” or “contour curves” on the plane.

• Method:

1. Set for various constant values of .

2. The equation represents a curve in the plane. Each curve corresponds to a specific height on the surface.

3. Plotting these curves together creates a contour map, which is a 2D representation of the 3D surface.

1. Set .

2. Plot for each : ∘ (Parabola opening upwards, vertex at (0, 2)) ∘ (Parabola opening upwards, vertex at (0, 1)) o (Parabola opening upwards, vertex at ( 0,0 )) ∘ (Parabola opening upwards, vertex at ( )) o (Parabola opening upwards, vertex at (0, -2))

Limits and Continuity of Multivariate Functions

Limits

• Definition: The limit of as ( ) approaches ( ) exists and equals if can be made arbitrarily close to by taking sufficiently close to , regardless of the path taken.

• Non-Existence: If different paths of approach yield different limits, the limit does not exist. This is often the easiest way to prove a limit doesn’t exist.

• Methods to Evaluate/Disprove Limits:

1. Direct Substitution: If the function is continuous at , simply substitute the values.

2. Orthogonal Lines (Freezing Variables): Evaluate the limit by setting one variable constant (e.g., or ) and taking the limit with respect to the other variable. If the limits obtained by freezing and freezing are different, the overall limit does not exist.

3. Parametrization (e.g., lines through the origin): Substitute or polar coordinates . If the resulting limit depends on or , the limit does not exist.

4. Testing Paths: Approach ( ) along different paths (e.g., ). If limits differ, the limit DNE.

1. Check Direct Substitution: , indeterminate.

2. Test Paths: o Path 1: Along the x-axis :

• Path 2: Along the axis :

Continuity

• Definition: A function is continuous at a point if:

1. is defined.

2. exists.

3. .

Partial Differentiation

Slope of a Surface

• Concept: For a surface , the “slope” at a point can be considered in different directions. The partial derivatives give the slope in the directions parallel to the coordinate axes.

• Partial Derivative with respect to ( or ): This is the slope of the surface in the direction of the x-axis. To calculate it, treat as a constant and differentiate with respect to .

• Partial Derivative with respect to ( or ): This is the slope of the surface in the direction of the y-axis. To calculate it, treat as a constant and differentiate with respect to .

Partial Differentiation by First Principles

• Formula:

Partial Differentiation by Fixing Variables

• Method: To find , treat all variables other than as constants and differentiate with respect to . Similarly, to find , treat all variables other than as constants and differentiate with respect to .

1. Partial derivative with respect to : Treat as a constant.

1. Partial derivative with respect to : Treat as a constant.

Second-Order Partial Derivatives

• Definition: Differentiating a first-order partial derivative again.

• Notation:

• Clairaut’s Theorem (Symmetry of Mixed Partials): If and are continuous in a region, then .

Tangent Plane

• Concept: The tangent plane is a plane that “touches” the surface at a specific point and has the same slope as the surface at that point in the and directions.

• Equation: The equation of the tangent plane at is:

1. Find . The point is .

2. Find partial derivatives:

3. Evaluate partial derivatives at

4. Plug into tangent plane equation:

Gradient

• Definition: The gradient of a scalar-valued function (like or ) is a vector that points in the direction of the greatest rate of increase of the function, and its magnitude is that greatest rate of increase.

• Notation: or .

• Formula (for ):

• Formula (for ):

• Magnitude: The magnitude represents the maximum rate of change.

• Direction: The direction of is the direction of the steepest ascent. The direction of is the direction of the steepest descent.

1. Find partial derivatives:

1. Evaluate at

2. Form the gradient vector:

Double Integrals

• Notation:

• dA: Represents an infinitesimal area element in the plane. In Cartesian coordinates, or .

• Evaluation (Iterated Integrals): Double integrals are typically evaluated as iterated integrals. The order of integration ( or ) can sometimes be swapped (Fubini’s Theorem), especially over rectangular regions. o Order

• The inner integral treats as a constant. The limits and can be functions of .

• The outer integral integrates the result of the inner integral with respect to . The limits and are constants.

• Order

• The inner integral treats as a constant. The limits and can be functions of .

• The outer integral integrates the result with respect to . The limits and are constants.

1. Set up as iterated integral: We can choose either order. Let’s use .

1. Evaluate inner integral (with respect to , treating as constant):

1. Evaluate outer integral (with respect to ):

1. Sketch the Region: The region is a triangle with vertices at .

2. Set up Iterated Integral: It’s usually easier to integrate for this shape.

• goes from 0 to 2 .

• For a fixed goes from the lower boundary ( ) to the upper boundary ( ).

1. Evaluate Inner Integral (with respect to , treating as constant):

1. Evaluate Outer Integral (with respect to ):