【AP Calculus】Integration Techniques & Applications (Past 10-Year FRQs)

Table of Contents

1. Basic Integration Methods
   • Power Rule & Exponential Functions
   • Trigonometric Integrals
   • Substitution Rule
   • Integration by Parts
   • Partial Fractions
   • Trigonometric Substitution
2. Definite Integral Applications
   • Area Between Curves
   • Volumes of Revolution
   • Arc Length
   • Average Value
3. Improper Integrals
4. Differential Equations
5. Physical Applications
6. FRQ Solutions

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1. Basic Integration Methods

1.1 Power Rule & Exponential Functions

Key Formulas

• $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
• $\int e^x \, dx = e^x + C$

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1.2 Trigonometric Integrals

Key Formulas

• $\int \sin x \, dx = -\cos x + C$
• $\int \sec^2 x \, dx = \tan x + C$
• $\int \tan x \, dx = \ln|\sec x| + C$

FRQ 2019 Q2(a)
Find $\int \frac{\sin x}{\cos^3 x} \, dx$.

Solution:

$$\int \frac{\sin x}{\cos^3 x} \, dx = \int \sin x \cdot \cos^{-3} x \, dx$$

Let $ u = \cos x $, $ du = -\sin x \, dx $:

$$= -\int u^{-3} \, du = -\left( \frac{u^{-2}}{-2} \right) + C = \frac{1}{2\cos^2 x} + C$$

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1.3 Substitution Rule

FRQ 2021 Q3(a)
Evaluate $\int_0^{\pi/4} \tan^3 x \sec^2 x \, dx$.

Solution:
Let $ u = \tan x $, $ du = \sec^2 x \, dx $.
When $ x = 0 $, $ u = 0 $; $ x = \pi/4 $, $ u = 1 $:

$$\int_0^1 u^3 \, du = \left[ \frac{u^4}{4} \right]_0^1 = \frac{1}{4}$$

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1.4 Integration by Parts

Formula: $\int u \, dv = uv - \int v \, du$

FRQ 2017 Q6(b)
Compute $\int x^2 \ln x \, dx$.

Solution:
Let $ u = \ln x $, $ dv = x^2 \, dx $:
$ du = \frac{1}{x} dx $, $ v = \frac{x^3}{3} $:

$$= \frac{x^3}{3} \ln x - \int \frac{x^3}{3} \cdot \frac{1}{x} dx = \frac{x^3}{3} \ln x - \frac{1}{3} \int x^2 dx$$ $$= \frac{x^3}{3} \ln x - \frac{x^3}{9} + C$$

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2. Definite Integral Applications

2.1 Area Between Curves

FRQ 2016 Q1
Let $ R $ be the region enclosed by $ y = 2x $ and $ y = x^2 $. Find the area of $ R $.

Solution:
Intersection points: $ 2x = x^2 \Rightarrow x = 0, 2 $:

$$\text{Area} = \int_0^2 (2x - x^2) \, dx = \left[ x^2 - \frac{x^3}{3} \right]_0^2 = 4 - \frac{8}{3} = \frac{4}{3}$$

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2.2 Volumes of Revolution (Disk Method)

FRQ 2019 Q3
Find the volume generated by rotating $ y = \sqrt{x} $ about the x-axis from $ x = 0 $ to $ x = 4 $.

Solution:

$$V = \pi \int_0^4 (\sqrt{x})^2 \, dx = \pi \int_0^4 x \, dx = \pi \left[ \frac{x^2}{2} \right]_0^4 = 8\pi$$

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2.3 Arc Length

FRQ 2021 Q2(c)
Calculate the length of $ y = \frac{2}{3}x^{3/2} $ from $ x = 0 $ to $ x = 3 $.

Solution:
$ f’(x) = x^{1/2} $:

$$L = \int_0^3 \sqrt{1 + x} \, dx = \left[ \frac{2}{3}(1 + x)^{3/2} \right]_0^3 = \frac{2}{3}(8 - 1) = \frac{14}{3}$$

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3. Improper Integrals

FRQ 2018 Q5
Determine if $\int_1^\infty \frac{1}{x^2} \, dx$ converges.

Solution:

$$\int_1^\infty \frac{1}{x^2} \, dx = \lim_{b \to \infty} \left[ -\frac{1}{x} \right]_1^b = \lim_{b \to \infty} \left( -\frac{1}{b} + 1 \right) = 1 \quad (\text{Converges})$$

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4. Differential Equations

FRQ 2022 Q4
Solve $\frac{dy}{dx} = 2x$ with $ y(0) = 1 $.

Solution:

$$y = \int 2x \, dx = x^2 + C$$

Apply $ y(0) = 1 $:

$$1 = 0 + C \Rightarrow C = 1 \quad \Rightarrow \quad y = x^2 + 1$$

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5. Physical Applications

Work Done

FRQ 2015 Q6
A force of $ F(x) = 3x^2 $ N moves an object along the x-axis from $ x = 1 $ to $ x = 3 $. Compute the work done.

Solution:

$$W = \int_1^3 3x^2 \, dx = \left[ x^3 \right]_1^3 = 27 - 1 = 26 \ \text{J}$$

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6. FRQ Solutions

FRQ 2020 Q2 (Modified)

Problem:
Let $ R $ be the region bounded by $ y = e^{-x} $, $ y = 0 $, $ x = 0 $, and $ x = 1 $. Find the volume when $ R $ is rotated about the y-axis.

Solution (Shell Method):

$$V = 2\pi \int_0^1 x \cdot e^{-x} \, dx$$

Using integration by parts:
Let $ u = x $, $ dv = e^{-x} dx $; $ du = dx $, $ v = -e^{-x} $:

$$= 2\pi \left[ -x e^{-x} \bigg|_0^1 + \int_0^1 e^{-x} dx \right] = 2\pi \left[ -e^{-1} + (-e^{-x}) \bigg|_0^1 \right] = 2\pi \left( 1 - \frac{2}{e} \right)$$

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💡 Exam Tips:

• Prioritize substitution when seeing composite functions (e.g., $\tan x \cdot \sec^2 x$).
• Disk vs. Shell Method: Disk for rotation around horizontal axis, Shell for vertical axis.
• Always check limits for improper integrals.
• Units matter in physics applications (e.g., Joules for work).