AP Course

AP Calculus AB/BC

Study the complete College Board sequence for AP Calculus AB and BC, from limits through infinite series.

Choose an official unit and topic to open its lecture, concept check, or focused practice.

Lessons

1.1 Introducing Calculus: Can Change Occur at an Instant?1.2 Defining Limits and Using Limit Notation1.3 Estimating Limit Values from Graphs1.4 Estimating Limit Values from Tables1.5 Determining Limits Using Algebraic Properties of Limits1.6 Determining Limits Using Algebraic Manipulation1.7 Selecting Procedures for Determining Limits1.8 Determining Limits Using the Squeeze Theorem1.9 Connecting Multiple Representations of Limits1.10 Exploring Types of Discontinuities1.11 Defining Continuity at a Point1.12 Confirming Continuity over an Interval1.13 Removing Discontinuities1.14 Connecting Infinite Limits and Vertical Asymptotes1.15 Connecting Limits at Infinity and Horizontal Asymptotes1.16 Working with the Intermediate Value Theorem (IVT)2.1 Defining Average and Instantaneous Rates of Change at a Point2.2 Defining the Derivative of a Function and Using Derivative Notation2.3 Estimating Derivatives of a Function at a Point2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist2.5 Applying the Power Rule2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple2.7 Derivatives of cos x, sin x, e^x, and ln x2.8 The Product Rule2.9 The Quotient Rule2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions3.1 The Chain Rule3.2 Implicit Differentiation3.3 Differentiating Inverse Functions3.4 Differentiating Inverse Trigonometric Functions3.5 Selecting Procedures for Calculating Derivatives3.6 Calculating Higher-Order Derivatives4.1 Interpreting the Meaning of the Derivative in Context4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration4.3 Rates of Change in Applied Contexts Other Than Motion4.4 Introduction to Related Rates4.5 Solving Related Rates Problems4.6 Approximating Values of a Function Using Local Linearity and Linearization4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms5.1 Using the Mean Value Theorem5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points5.3 Determining Intervals on Which a Function Is Increasing or Decreasing5.4 Using the First Derivative Test to Determine Relative (Local) Extrema5.5 Using the Candidates Test to Determine Absolute (Global) Extrema5.6 Determining Concavity of Functions over Their Domains5.7 Using the Second Derivative Test to Determine Extrema5.8 Sketching Graphs of Functions and Their Derivatives5.9 Connecting a Function, Its First Derivative, and Its Second Derivative5.10 Introduction to Optimization Problems5.11 Solving Optimization Problems5.12 Exploring Behaviors of Implicit Relations6.1 Exploring Accumulations of Change6.2 Approximating Areas with Riemann Sums6.3 Riemann Sums, Summation Notation, and Definite Integral Notation6.4 The Fundamental Theorem of Calculus and Accumulation Functions6.5 Interpreting the Behavior of Accumulation Functions Involving Area6.6 Applying Properties of Definite Integrals6.7 The Fundamental Theorem of Calculus and Definite Integrals6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation6.9 Integrating Using Substitution6.10 Integrating Functions Using Long Division and Completing the Square6.11 Integrating Using Integration by Parts6.12 Using Linear Partial Fractions6.13 Evaluating Improper Integrals6.14 Selecting Techniques for Antidifferentiation7.1 Modeling Situations with Differential Equations7.2 Verifying Solutions for Differential Equations7.3 Sketching Slope Fields7.4 Reasoning Using Slope Fields7.5 Approximating Solutions Using Euler’s Method7.6 Finding General Solutions Using Separation of Variables7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables7.8 Exponential Models with Differential Equations7.9 Logistic Models with Differential Equations8.1 Finding the Average Value of a Function on an Interval8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts8.4 Finding the Area Between Curves Expressed as Functions of x8.5 Finding the Area Between Curves Expressed as Functions of y8.6 Finding the Area Between Curves That Intersect at More Than Two Points8.7 Volumes with Cross Sections: Squares and Rectangles8.8 Volumes with Cross Sections: Triangles and Semicircles8.9 Volume with Disc Method: Revolving Around the x- or y-Axis8.10 Volume with Disc Method: Revolving Around Other Axes8.11 Volume with Washer Method: Revolving Around the x- or y-Axis8.12 Volume with Washer Method: Revolving Around Other Axes8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled9.1 Defining and Differentiating Parametric Equations9.2 Second Derivatives of Parametric Equations9.3 Finding Arc Lengths of Curves Given by Parametric Equations9.4 Defining and Differentiating Vector-Valued Functions9.5 Integrating Vector-Valued Functions9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions9.7 Defining Polar Coordinates and Differentiating in Polar Form9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve9.9 Finding the Area of the Region Bounded by Two Polar Curves10.1 Defining Convergent and Divergent Infinite Series10.2 Working with Geometric Series10.3 The nth Term Test for Divergence10.4 Integral Test for Convergence10.5 Harmonic Series and p-Series10.6 Comparison Tests for Convergence10.7 Alternating Series Test for Convergence10.8 Ratio Test for Convergence10.9 Determining Absolute or Conditional Convergence10.10 Alternating Series Error Bound10.11 Finding Taylor Polynomial Approximations of Functions10.12 Lagrange Error Bound10.13 Radius and Interval of Convergence of Power Series10.14 Finding Taylor or Maclaurin Series for a Function10.15 Representing Functions as Power Series

Quizzes

Practice Problems AP formula notes, graph references, and practice sets will be added here.

AP Calculus AB/BC · Unit 6 · Topic 6.13 · BC Only

Evaluating Improper Integrals

Replace infinite bounds or unbounded integrands with limits and test convergence.

1. Topic Focus

Interpret definite integrals as accumulated change, connect sums to integrals, apply both Fundamental Theorems, and select antiderivative techniques.

This topic: Replace infinite bounds or unbounded integrands with limits and test convergence.

2. Key Relationship

\(\int_a^\infty f=\lim\limits_{b\to\infty}\int_a^b f\)

Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.

3. Visual Connection

Replace every improper feature with its own limitAn infinite tail needs a moving finite bound, while an interior pole needs two independent one-sided limits.

4. Worked Example

The integral ∫₁^∞ 1/x² dx converges to 1, while ∫₁^∞ 1/x dx diverges.

Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.

5. Concept Development

1. Improper integrals are limit problems

An ordinary definite integral assumes a finite interval and a bounded integrand. If either condition fails, the integral is improper. Its notation is shorthand for one or more limits of proper definite integrals. The integral converges only when every required limit exists as a finite real number.

2. Recognize the two sources of impropriety

Infinite interval: at least one bound is \(\infty\) or \(-\infty\).
Unbounded integrand: the function grows without bound at an endpoint or at a point inside the interval.

Inspect both the bounds and the domain of the integrand before using the Fundamental Theorem of Calculus.

3. Infinity is not an endpoint value

The symbol \(\infty\) describes unbounded behavior; it is not a number that can be substituted into an antiderivative. Introduce a finite variable first, evaluate the proper integral, and only then take a limit.

4. Infinite upper bound

\(\int_a^\infty f(x)\,dx=\lim\limits_{b\to\infty}\int_a^b f(x)\,dx.\)

If the limit is finite, that limit is the value of the improper integral. If it is infinite or does not exist, the integral diverges.

5. Infinite lower bound

\(\int_{-\infty}^b f(x)\,dx=\lim\limits_{a\to-\infty}\int_a^b f(x)\,dx.\)

The replacement variable approaches \(-\infty\) from within the interval. The same finite-limit requirement applies.

6. Both bounds infinite

Choose any convenient finite number \(c\), usually \(0\), and test the two sides independently:

\(\int_{-\infty}^{\infty}f(x)\,dx=\int_{-\infty}^{c}f(x)\,dx+\int_c^{\infty}f(x)\,dx.\)

The whole integral converges only if both improper integrals converge. A single symmetric limit from \(-b\) to \(b\) is not the definition of ordinary convergence.

7. Unbounded at an endpoint

If \(f\) is unbounded as \(x\to a^+\), approach the endpoint from inside the interval:

\(\int_a^b f(x)\,dx=\lim\limits_{t\to a^+}\int_t^b f(x)\,dx.\)

For a singularity at \(b\), use \(t\to b^-\). The direction on the limit is part of the definition.

8. An interior singularity requires a split

If \(f\) is unbounded at \(x=c\) with \(a<c<b\), create two one-sided integrals:

\(\int_a^b f(x)\,dx=\lim\limits_{t\to c^-}\int_a^t f(x)\,dx+\lim\limits_{s\to c^+}\int_s^b f(x)\,dx.\)

Do not integrate across the singularity in one step. Each side must produce a finite limit.

9. Every improper piece must converge

A finite result on one side cannot rescue divergence on another. Once any required component diverges, the original integral diverges and the remaining pieces need not be combined.

10. The tail \(p\)-integral benchmark

\(\int_1^\infty \frac{1}{x^p}\,dx\text{ converges exactly when }p>1.\)

For \(p>1\), the tail decays fast enough to have finite accumulated area. For \(p\le 1\), it diverges.

11. The near-zero \(p\)-integral benchmark

\(\int_0^1 \frac{1}{x^p}\,dx\text{ converges exactly when }p<1.\)

The inequality reverses because the question is now how sharply the function blows up near zero, not how quickly it decays at infinity.

12. Why \(p=1\) is the boundary

Both \(\int_1^\infty 1/x\,dx\) and \(\int_0^1 1/x\,dx\) produce logarithms whose magnitudes grow without bound. Memorize the two \(p\)-rules, but connect their common boundary to logarithmic divergence.

13. A reliable evaluation workflow

Locate every infinite bound and every point where the integrand is unbounded.
Split the interval at each improper point.
Write each piece as a correctly directed limit.
Evaluate the finite definite integral before taking the limit.
State converges to ... or diverges, with the limiting evidence.

14. Fast decay can produce finite accumulation

An interval may have infinite length while its integral is finite. Negative exponential functions and powers \(x^{-p}\) with \(p>1\) are standard examples because their values decrease rapidly enough.

15. A graph guides setup, not proof

A graph can reveal an infinite tail, a vertical asymptote, or a hidden domain break. It also helps check whether a positive-area answer should be finite or infinite. The convergence conclusion, however, must come from the required limits.

16. Symmetric cancellation is not ordinary convergence

For an odd function, \(\int_{-b}^{b}f(x)\,dx\) may equal zero for every \(b\). The improper integral from \(-\infty\) to \(\infty\) still diverges if either one-sided integral diverges. Such symmetric cancellation is a different idea called a principal value.

17. Keep signs and area interpretations separate

An improper integral is signed accumulation. If \(f\ge0\), convergence means the unbounded region has finite area. If \(f\) changes sign, a finite integral is a net value; ordinary convergence still requires all separately defined improper pieces to converge.

18. Final error check

Do not write \(F(\infty)\), ignore a vertical asymptote, combine \(\infty-\infty\), or stop after finding an antiderivative. The limits are the calculation, and a complete answer names convergence or divergence explicitly.

6. Detailed Worked Example and Error Check

Example 1: A convergent infinite tail.

\(\begin{aligned}\int_2^\infty \frac{1}{x^3}\,dx&=\lim\limits_{b\to\infty}\int_2^b x^{-3}\,dx\\&=\lim\limits_{b\to\infty}\left[-\frac{1}{2x^2}\right]_2^b=\lim\limits_{b\to\infty}\left(\frac18-\frac{1}{2b^2}\right)=\frac18.\end{aligned}\)

The limit is finite, so the integral converges to \(1/8\).

Example 2: A logarithmically divergent tail.

\(\int_3^\infty\frac{1}{x}\,dx=\lim\limits_{b\to\infty}(\ln b-\ln3)=\infty.\)

The integral diverges. The curve approaches zero, but approaching zero alone does not guarantee finite accumulated area.

Example 3: A convergent endpoint singularity.

\(\begin{aligned}\int_0^4\frac{1}{\sqrt{x}}\,dx&=\lim\limits_{a\to0^+}\int_a^4x^{-1/2}\,dx\\&=\lim\limits_{a\to0^+}[2\sqrt{x}]_a^4=\lim\limits_{a\to0^+}(4-2\sqrt a)=4.\end{aligned}\)

Although the integrand is unbounded at zero, the improper integral converges.

Example 4: A divergent endpoint singularity.

\(\int_0^1x^{-3/2}\,dx=\lim\limits_{a\to0^+}\left[-2x^{-1/2}\right]_a^1=\lim\limits_{a\to0^+}\left(-2+\frac{2}{\sqrt a}\right)=\infty.\)

Here \(p=3/2>1\), so the near-zero \(p\)-integral rule also predicts divergence.

Example 5: An interior vertical asymptote.

For \(\int_0^3 1/(x-1)^2\,dx\), split at \(x=1\). On the left,

\(\lim\limits_{t\to1^-}\int_0^t\frac{dx}{(x-1)^2}=\lim\limits_{t\to1^-}\left[-\frac{1}{x-1}\right]_0^t=\infty.\)

That one divergent piece is enough to conclude that the original integral diverges.

Example 6: Both bounds infinite.

\(\begin{aligned}\int_{-\infty}^{\infty}\frac{dx}{1+x^2}&=\int_{-\infty}^{0}\frac{dx}{1+x^2}+\int_0^{\infty}\frac{dx}{1+x^2}\\&=\frac{\pi}{2}+\frac{\pi}{2}=\pi.\end{aligned}\)

Each value follows from a separate one-sided limit of \(\arctan x\); therefore the entire integral converges.

Example 7: Exponential decay on an infinite interval.

\(\int_0^\infty e^{-2x}\,dx=\lim\limits_{b\to\infty}\left[-\frac12e^{-2x}\right]_0^b=\frac12.\)

The positive area is finite because \(e^{-2b}\to0\).

7. AP Reasoning Routine

Identify the accumulating quantity and units, preserve bounds, choose a valid integration technique, and check answers by differentiation.

Identify the representation and requested quantity.
State the rule or theorem and verify its conditions.
Keep exact values until the final requested approximation.
Interpret sign, units, interval, and context.

AP Checkpoint

Evaluate each improper integral or state that it diverges. Show every required limit.
(a) \(\int_1^\infty x^{-4}\,dx\).
(b) \(\int_4^\infty x^{-1/2}\,dx\).
(c) \(\int_0^8 x^{-2/3}\,dx\).
(d) \(\int_0^1 x^{-5/4}\,dx\).
(e) \(\int_{-\infty}^{0}e^{3x}\,dx\).
(f) \(\int_{-\infty}^{\infty}e^{-|x|}\,dx\).
(g) \(\int_{-1}^{2}\frac{1}{(x-1)^2}\,dx\).
(h) \(\int_0^2\frac{1}{\sqrt{2-x}}\,dx\).
(i) Explain why \(\int_{-\infty}^{\infty}x\,dx\) is not zero even though \(\int_{-b}^{b}x\,dx=0\).
(j) Give the convergence conditions for \(\int_1^\infty x^{-p}\,dx\) and \(\int_0^1x^{-p}\,dx\), and explain the role of \(p=1\).

Check the solution

(a) \(\lim\limits_{b\to\infty}[-1/(3x^3)]_1^b=1/3\), so it converges to \(1/3\).
(b) \(\lim\limits_{b\to\infty}[2\sqrt{x}]_4^b=\lim\limits_{b\to\infty}(2\sqrt b-4)=\infty\); it diverges.
(c) \(\lim\limits_{a\to0^+}[3x^{1/3}]_a^8=6\), so it converges to \(6\).
(d) \(\lim\limits_{a\to0^+}[-4x^{-1/4}]_a^1=\lim\limits_{a\to0^+}(-4+4a^{-1/4})=\infty\); it diverges.
(e) \(\lim\limits_{a\to-\infty}[e^{3x}/3]_a^0=1/3\), so it converges to \(1/3\).
(f) Split at zero. Each half equals \(1\): on the left integrate \(e^x\), and on the right integrate \(e^{-x}\). Both converge, so the total is \(2\).
(g) The integrand is unbounded at the interior point \(x=1\). The left-hand integral already tends to \(\infty\), so the original integral diverges.
(h) \(\lim\limits_{b\to2^-}[-2\sqrt{2-x}]_0^b=2\sqrt2\), so it converges to \(2\sqrt2\).
(i) Ordinary convergence requires separate integrals on \(( -\infty,0]\) and \([0,\infty)\). They diverge to \(-\infty\) and \(\infty\), respectively, so their values cannot be canceled. The symmetric value zero is only a principal value.
(j) The tail integral converges for \(p>1\); the near-zero integral converges for \(p<1\). At \(p=1\), both reduce to logarithmic limits and diverge.