AP Calculus AB Sections 3.13.4 Study Guide & Practice Test

Why Sections 3.13.4 Matter

Sections 3.1 through 3.4 cover the foundations of integration, an essential component of the AP Calculus AB curriculum. Mastery of these sections prepares you for:

• Understanding the definite integral as area under a curve.
• Applying the Fundamental Theorem of Calculus.
• Evaluating integrals using substitution and basic antiderivatives.
• Solving applied problems involving accumulation, velocity, and average value.

Because the AP exam frequently tests these concepts in both multiplechoice and freeresponse formats, a focused study guide and targeted practice test can boost your confidence and score.

Section 3.1 Riemann Sums and Approximation

Key Ideas

Common Mistakes

• Confusing x with the width of an individual subinterval when the partition isnt uniform.
• Forgetting to adjust the limits of summation when switching from left to right endpoints.
• Neglecting absolute values when f(x) becomes negative on part of the interval.

Study Tips

• Practice converting a given sum into an integral and viceversa.
• Use a graphing calculator or software to visualize how the sum approaches the true area as n increases.
• Memorize the formula for a midpoint Riemann sum: f( (x_i1 + x_i) / 2 ) x.

Section 3.2 The Definite Integral

Key Ideas

Typical Exam Questions

• Compute the area between two curves using (top bottom) dx.
• Determine the net change of a quantity described by its rate function.
• Apply the properties above to simplify an expression before evaluating.

Preparation Strategy

• Work through at least three problems that require setting up integrals from word problems.
• Review how to split a region into multiple integrals when the integrand changes sign.
• Practice using symmetry to simplify calculations (e.g., even/odd functions).

Section 3.3 The Fundamental Theorem of Calculus (FTC)

Part 1 (Evaluation)

Part 2 (Differentiation)

Common Pitfalls

• Using the FTC on an integral where the integrand is not continuous on the interval.
• Confusing the variable of integration (t) with the variable of differentiation (x).
• Forgetting to multiply by the derivative of the inner function when the limit is g(x).

Tips for Mastery

• Identify the antiderivative before applying the FTC; practice with a variety of functions (polynomials, trig, exponential).
• Set up a quick FTC checklist before solving a problem: continuity? correct limits? correct antiderivative?
• Do timed drills that require you to evaluate integrals in under a minute.

Section 3.4 Integration Techniques (Basic Antiderivatives)

Core Antiderivatives

Simple Substitution

When the integrand contains a function and its derivative, use usubstitution. The steps are:

1. Identify u = g(x) whose derivative appears elsewhere in the integrand.
2. Rewrite the integral in terms of du.
3. Integrate, then substitute back.

Typical Test Items

• Integrate functions like (3x + 2x) e ^{x + x} using u = x + x.
• Find the area between curves that require a substitution for the antiderivative.
• Evaluate definite integrals where the limits change after substitution.

Practice Recommendations

• Complete a set of at least ten substitution problems, mixing both indefinite and definite integrals.
• Review how to adjust the limits of integration when the substitution changes the variable.
• Use quick pattern matching to recognize when a function fits a standard form (e.g., derivative of denominator appears in numerator).

Practice Test Putting It All Together

MultipleChoice Questions

1. For f(x)=x6x, the leftendpoint Riemann sum with n=4 on [0,2] is:
   • A) 4
   • B) 6
   • C) 8
   • D) 10
2. The definite integral ₁⁴ (3t) dt equals:
   • A) 9
   • B) 12
   • C) 15
   • D) 21
3. Given G(x)=₀^x (sin t) dt, G'(/2) equals:
   • A) 0
   • B)
   • C) 1
   • D) undefined
4. Use substitution to evaluate (2x) e^x dx. The antiderivative is:
   • A) e^x + C
   • B) e^x + C
   • C) 2e^x + C
   • D) x e^x + C
5. The area between y = x and y = 4 on [2,2] is:
   • A) 8/3
   • B) 16/3
   • C) 32/3
   • D) 64/3
6. Apply the FTC Part2 to find d/dx ₀^{cos x} (1+t) dt:
   • A) (1+cosx)sin x
   • B) (1+cosx)sin x
   • C) (1+cosx)
   • D) (1+cosx)
7. (5 / x) dx equals:
   • A) 5 ln|x| + C
   • B) 5x + C
   • C) ln|5x| + C
   • D) (5/x) + C
8. For f(x)=e^x, the midpoint Riemann sum with n=2 on [0,2] is:
   • A) 2/e
   • B) e1/e
   • C) 11/e
   • D) e1

FreeResponse Prompts

9. Let f(x)=x3x. Use a Riemann sum to set up an integral that represents the net area between the curve and the xaxis on the interval [1,1]. Then evaluate the integral.

10. Define G(x)=₂^x (1/t) dt. Find G'(3) using the Fundamental Theorem of Calculus and the chain rule.

Solution Key (Brief)

• 1. B) 6 compute left endpoints x=0,0.5,1,1.5 and sum f(x)x.
• 2. D) 21 antiderivative x evaluated from 1 to 4 gives 641=63; multiply by 3 gives 21.
• 3. C) 1 G'(x)=sin x, so G'(/2)=sin(/2)=1.
• 4. A) e^x + C set u=x, du=2x dx.
• 5. B) 16/3 area = ₂² (4x) dx = [4xx/3] from 2 to 2 = 16/3.
• 6. A) (1+cosx)sin x differentiate using the chain rule.
• 7. A) 5 ln|x| + C standard logarithmic antiderivative.
• 8. C) 11/e midpoint at x=1 gives e¹, x=2, sum =2e¹=2/e; subtract from total area 2 gives 11/e.
• 9. Integral: ₁¹ (x3x) dx = 0 (odd function); net area = 0.
• 10. G'(x)= (1/(x))2x = 2/x; thus G'(3)=2/3.

Additional Resources

To deepen your understanding, explore these free online tools:

• Khan Academy AP Calculus AB
• Desmos Graphing Calculator (great for visualizing Riemann sums)
• College Board AP Calculus AB Course Description
• OpenStax Calculus Volume 1 (Chapters 45 cover integration)