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Integration Applications

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Integration Applications - Complete Interactive Lesson

Part 1: Area Between Curves (Advanced)

Integration Applications

Part 1 of 7 — Area Between Curves

In This Topic

Part	Topic
1	Area Between Curves
2	Cross-Sectional Volumes
3	Disk & Washer Methods
4	Riemann Sums & Trapezoidal Rule
5	Rate Problems & Net Change
6	Problem-Solving Workshop
7	Comprehensive Assessment

Area Between Two Curves

$\boxed{A = \int_a^b [f(x) - g(x)]\,dx \quad \text{where } f(x) \ge g(x)}$

Step	Action
1	Sketch and identify which curve is on top
2	Find intersection points (set $f = g$ )
3	Set up $\int [\text{top} - \text{bottom}]\,dx$
4	If curves cross, split at the crossing points

Key Fact: Always subtract bottom from top: $\int (\text{top} - \text{bottom})\,dx$ . If you get a negative area, the curves were swapped.

Worked Examples

Example 1: Area between $y = x^2$ and $y = x$ on $[0,1]$ .

Step	Work
Intersections	$x^2 = x \Rightarrow x=0,1$
Top curve	$x \ge x^2$ on $[0,1]$
Integral	$\int_0^1 (x-x^2)dx = [\frac{x^2}{2}-\frac{x^3}{3}]_0^1$
Answer	$\frac{1}{2}-\frac{1}{3} = \frac{1}{6}$

Example 2: Area enclosed by $y = x^2$ and $y = 2x$ .

Step	Work
Intersections	$x^2 = 2x \Rightarrow x=0,2$
Top curve	$2x \ge x^2$ on $[0,2]$
Integral	$\int_0^2(2x-x^2)dx = [x^2-\frac{x^3}{3}]_0^2$
Answer	$4-\frac{8}{3} = \frac{4}{3}$

Integrating with Respect to $y$

$\boxed{A = \int_c^d [\text{right} - \text{left}]\,dy}$

Use when curves are easier to express as $x = f(y)$ .

AP Tip: If the region is bounded by $x = y^2$ and $x = 4$ , integrate with respect to $y$ . It avoids splitting the integral.

Area Between Curves 🎯

Set up the integral. 🔍

Compute the area. ✍️

Key Takeaways — Part 1

Concept	Formula
Area (horizontal)	$\int_a^b [\text{top}-\text{bottom}]\,dx$
Area (vertical)	$\int_c^d [\text{right}-\text{left}]\,dy$
Crossing curves	Split at intersection points

Up Next: Part 2 — Cross-Sectional Volumes.

Part 2: Cross-Sectional Volumes

Integration Applications

Part 2 of 7 — Cross-Sectional Volumes

Volume with Known Cross Sections

$\boxed{V = \int_a^b A(x)\,dx}$

where $A(x)$ is the area of the cross section at position $x$ .

Cross-Section Area Formulas

If the side length (or diameter) of each cross section is $s(x) = f(x) - g(x)$ :

Shape	Area $A(x)$	Common AP form
Square	$s^2$	$(f(x)-g(x))^2$
Semicircle	$\frac{\pi}{8}s^2$	$\frac{\pi}{8}(f(x)-g(x))^2$
Equilateral triangle	$\frac{\sqrt{3}}{4}s^2$	$\frac{\sqrt{3}}{4}(f(x)-g(x))^2$
Isosceles right triangle (leg)	$\frac{1}{2}s^2$	$\frac{1}{2}(f(x)-g(x))^2$
Rectangle ( $h = 2s$ )	$2s^2$	$2(f(x)-g(x))^2$

Key Fact: The shape of the cross section only affects the constant multiplier. The integral setup is always $\int_a^b (\text{constant}) \cdot s^2\,dx$ .

Worked Example

Base region: Between $y = \sqrt{x}$ and $y = 0$ from $x=0$ to $x=4$ .

Cross sections (perpendicular to $x$ -axis) are squares.

Step	Work
Side length	$s(x) = \sqrt{x} - 0 = \sqrt{x}$
Cross-section area	$A(x) = (\sqrt{x})^2 = x$
Volume integral	$V = \int_0^4 x\,dx$
Evaluate	$[\frac{x^2}{2}]_0^4 = 8$

Same base, semicircular cross sections:

Step	Work
Diameter	$d = \sqrt{x}$ , radius $= \sqrt{x}/2$
Area	$\frac{\pi}{2}(\sqrt{x}/2)^2 = \frac{\pi x}{8}$
Volume	$\int_0^4 \frac{\pi x}{8}dx = \frac{\pi}{8} \cdot 8 = \pi$

AP Tip: Cross-section volumes are a favorite AP FRQ topic. Make sure you can set up the integral for any shape.

Cross-Sectional Volumes 🎯

Set up the volume integral. 🔍

Compute the volume. ✍️

Key Takeaways — Part 2

Shape	Area multiplier
Square	$1 \cdot s^2$
Semicircle	$\frac{\pi}{8}s^2$
Equilateral $\triangle$	$\frac{\sqrt{3}}{4}s^2$
Isosceles right $\triangle$	$\frac{1}{2}s^2$

Up Next: Part 3 — Disk & Washer Methods.

Part 3: Volumes: Disk and Washer Methods

Integration Applications

Part 3 of 7 — Disk & Washer Methods

Disk Method (Solid with No Hole)

$\boxed{V = \pi\int_a^b [R(x)]^2\,dx}$

Use when: The region is rotated around an axis and there is no gap between the region and the axis.

Washer Method (Solid with a Hole)

$\boxed{V = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx}$

Variable	Meaning
$R(x)$	Outer radius (farther curve from axis)
$r(x)$	Inner radius (closer curve to axis)

Key Fact: NEVER subtract the radii first. It’s $R^2 - r^2$ , NOT $(R-r)^2$ .

Rotation About Different Lines

Axis of Rotation	Radius Setup
$x$ -axis ( $y=0$ )	$R = f(x)$
$y$ -axis ( $x=0$ )	$R = f(y)$ , integrate $dy$
$y = k$ (horizontal)	$R =
$x = h$ (vertical)	$R =

Worked Example

Rotate $y = x^2$ about the $x$ -axis from $x=0$ to $x=2$ :

Step	Work
Radius	$R = x^2$
Integral	$V = \pi\int_0^2 (x^2)^2 dx = \pi\int_0^2 x^4\,dx$
Evaluate	$\pi[\frac{x^5}{5}]_0^2 = \frac{32\pi}{5}$

Washer Example

Rotate region between $y=x$ and $y=x^2$ about $x$ -axis on $[0,1]$ :

$R = x$ (outer), $r = x^2$ (inner).

$V = \pi\int_0^1(x^2-x^4)dx = \pi[\frac{x^3}{3}-\frac{x^5}{5}]_0^1 = \pi(\frac{1}{3}-\frac{1}{5}) = \frac{2\pi}{15}$

Disk & Washer 🎯

Identify the setup. 🔍

Compute the volume. ✍️

Key Takeaways — Part 3

Method	When to Use	Formula
Disk	No gap from axis	$\pi\int R^2\,dx$
Washer	Gap creates hole	$\pi\int (R^2-r^2)\,dx$

Up Next: Part 4 — Riemann Sums & Trapezoidal Rule.

Part 4: Riemann Sums and Trapezoidal Rule

Integration Applications

Part 4 of 7 — Riemann Sums & Trapezoidal Rule

Riemann Sum Formulas

$\boxed{L_n = \sum_{i=0}^{n-1} f(x_i)\Delta x \qquad R_n = \sum_{i=1}^{n} f(x_i)\Delta x \qquad M_n = \sum_{i=1}^{n} f(\bar{x}_i)\Delta x}$

Trapezoidal Rule

$\boxed{T_n = \frac{\Delta x}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]}$

Over/Underestimate Guide

Method	$f$ Increasing	$f$ Decreasing
Left	Under	Over
Right	Over	Under
Midpoint	Depends on concavity	Depends on concavity

Method	$f$ Concave Up	$f$ Concave Down
Trapezoidal	Over	Under
Midpoint	Under	Over

Key Fact: Trapezoidal = average of Left and Right: $T_n = \frac{L_n + R_n}{2}$ .

Worked Example from Table

$x$	$0$	$1$	$2$	$3$
$f(x)$	$1$	$3$	$2$	$5$

$\Delta x = 1$ , $n = 3$ .

Method	Computation	Value
Left sum	$f(0)+f(1)+f(2) = 1+3+2$	$6$
Right sum	$f(1)+f(2)+f(3) = 3+2+5$	$10$
Trapezoidal	$\frac{1}{2}(1+2(3)+2(2)+5) = \frac{16}{2}$	$8$

Check: $T = \frac{L+R}{2} = \frac{6+10}{2} = 8$ ✔

Unequal Subintervals

When $\Delta x$ varies, use individual trapezoids:

$T = \frac{\Delta x_1}{2}(f_0+f_1) + \frac{\Delta x_2}{2}(f_1+f_2) + \cdots$

AP Tip: Table problems with unequal spacing appear frequently. Apply the trapezoidal formula to each subinterval separately.

Riemann Sums 🎯

Over or under? 🔍

Compute from a table. ✍️

Key Takeaways — Part 4

Method	Formula
Left	$\sum f(x_i)\Delta x$ (skip last)
Right	$\sum f(x_i)\Delta x$ (skip first)
Trapezoidal	$\frac{\Delta x}{2}[f_0+2f_1+\cdots+2f_{n-1}+f_n]$
Key relation	$T = (L+R)/2$

Up Next: Part 5 — Rate Problems & Net Change.

Part 5: Rate Problems & Net Change

Integration Applications

Part 5 of 7 — Rate Problems & Net Change

The Net Change Theorem

$\boxed{\int_a^b f'(x)\,dx = f(b) - f(a)}$

The integral of a rate of change gives the net change in the quantity.

Common AP Contexts

Rate Function	What $\int$ Gives	Units
$v(t)$ (velocity)	Displacement	distance
$	v(t)	$ (speed)
$R(t)$ (flow rate)	Net volume change	volume
$P'(t)$ (pop. rate)	Net population change	count
$C'(x)$ (marginal cost)	Total cost change	dollars

Key Fact: "Rate" in the problem statement means you’re given $f'$ , and integration gives $f(b)-f(a)$ .

Rate In / Rate Out

$\boxed{Q(t) = Q_0 + \int_0^t [R_{in}(s) - R_{out}(s)]\,ds}$

AP FRQ Pattern

Part	Typical Question
(a)	Compute $\int_a^b R(t)\,dt$ and interpret
(b)	Is quantity increasing or decreasing at $t = c$ ?
(c)	Find absolute min/max on interval
(d)	Average rate over interval

Interpretation sentence template:

" $\int_a^b R(t)\,dt = N$ means that $N$ [units] of [quantity] [entered/left] from $t=a$ to $t=b$ ."

AP Tip: The AP exam almost always asks you to interpret the integral in context. Include units and the time interval.

Rate Problems 🎯

Interpret the integral. 🔍

Apply net change. ✍️

Key Takeaways — Part 5

Concept	Formula
Net change	$\int_a^b f'(t)\,dt = f(b)-f(a)$
Rate in/out	$Q_0 + \int_0^t(R_{in}-R_{out})ds$
Interpretation	Include units and time interval

Up Next: Part 6 — Problem-Solving Workshop.

Part 6: Practice Workshop

Integration Applications

Part 6 of 7 — Problem-Solving Workshop

Integration Application Decision Guide

Given	Method
Two curves, find area	$\int [\text{top}-\text{bottom}]dx$
Region + cross-section shape	$\int A(x)\,dx$ with shape formula
Rotation about axis, one boundary	Disk: $\pi\int R^2\,dx$
Rotation about axis, two boundaries	Washer: $\pi\int(R^2-r^2)dx$
Table of values	Riemann sum or trapezoidal rule
Rate function	Net change: $\int_a^b f'(t)\,dt$
Rate in / rate out	$Q_0 + \int(R_{in}-R_{out})dt$

Key Fact: The first step is always identifying the problem type. The correct setup determines 90% of your score.

AP-Style Worked Problem

Region $R$ is bounded by $y = 4-x^2$ and $y = 0$ .

(a) Find the area of $R$ .

$A = \int_{-2}^{2}(4-x^2)dx = [4x-\frac{x^3}{3}]_{-2}^{2} = \frac{32}{3}$

(b) Cross sections perpendicular to $x$ -axis are squares. Find volume.

$s = 4-x^2$ . $V = \int_{-2}^{2}(4-x^2)^2 dx$

$= \int_{-2}^{2}(16-8x^2+x^4)dx = 2[16x-\frac{8x^3}{3}+\frac{x^5}{5}]_0^2$

$= 2(32-\frac{64}{3}+\frac{32}{5}) = 2 \cdot \frac{480-320+96}{15} = \frac{512}{15}$

(c) Rotate $R$ about the $x$ -axis. Find volume.

$V = \pi\int_{-2}^{2}(4-x^2)^2 dx = \frac{512\pi}{15}$

AP Tip: Parts (b) and (c) have the same integral! Cross sections → no $\pi$ . Revolution → multiply by $\pi$ .

Mixed Applications 🎯

Choose the right setup. 🔍

Mixed problem. ✍️

Key Takeaways — Part 6

Problem Type	Key Setup
Area	$\int(\text{top}-\text{bottom})$
Cross-section	$\int A(x)\,dx$
Revolution	$\pi\int R^2$ or $\pi\int(R^2-r^2)$
Table data	Riemann sums or trapezoidal
Rates	$\int f'(t)\,dt = f(b)-f(a)$

Up Next: Part 7 — Comprehensive Assessment.

Part 7: Final Assessment

Integration Applications

Part 7 of 7 — Comprehensive Assessment

Complete Formula Reference

Application	Formula
Area (horizontal)	$\int_a^b[f(x)-g(x)]dx$
Area (vertical)	$\int_c^d[\text{right}-\text{left}]dy$
Cross-section volume	$\int_a^b A(x)\,dx$
Disk method	$\pi\int_a^b R^2\,dx$
Washer method	$\pi\int_a^b(R^2-r^2)dx$
Trapezoidal rule	$\frac{\Delta x}{2}[f_0+2f_1+\cdots+f_n]$
Net change	$\int_a^b f'(t)\,dt = f(b)-f(a)$
Average value	$\frac{1}{b-a}\int_a^b f(x)\,dx$

Top AP Mistakes

Mistake	Correction
Subtracting radii: $(R-r)^2$	Use $R^2-r^2$ in washer method
Forgetting $\pi$ in revolution	Cross-section: no $\pi$ . Revolution: include $\pi$ .
Wrong over/underestimate	Check increasing/decreasing AND concavity
Missing intersection points	Always find where curves cross
Wrong axis of rotation	Adjust radii: distance = $
Not interpreting with units	Always state quantity, units, and time interval

Quiz — Area & Volume 🎯

Quiz — Numerical & Rates 🎯

Final classification. 🔍

Final Challenge ✍️

Integration Applications — Complete!

You’ve mastered:

Part	Topic
1	Area between curves
2	Cross-sectional volumes
3	Disk & washer methods
4	Riemann sums & trapezoidal rule
5	Rate problems & net change
6	Problem-solving workshop
7	Comprehensive assessment

You’re ready for AP-level integration application problems!

← Back to Standard Lesson

Integration Applications

Integration Applications - Complete Interactive Lesson

Part 1: Area Between Curves (Advanced)

Integration Applications

In This Topic

Area Between Two Curves

Worked Examples

Integrating with Respect to yyy

Key Takeaways — Part 1

Part 2: Cross-Sectional Volumes

Integration Applications

Volume with Known Cross Sections

Cross-Section Area Formulas

Worked Example

Key Takeaways — Part 2

Part 3: Volumes: Disk and Washer Methods

Integration Applications

Disk Method (Solid with No Hole)

Washer Method (Solid with a Hole)

Rotation About Different Lines

Worked Example

Washer Example

Key Takeaways — Part 3

Part 4: Riemann Sums and Trapezoidal Rule

Integration Applications

Riemann Sum Formulas

Trapezoidal Rule

Over/Underestimate Guide

Worked Example from Table

Unequal Subintervals

Key Takeaways — Part 4

Part 5: Rate Problems & Net Change

Integration Applications

The Net Change Theorem

Common AP Contexts

Rate In / Rate Out

AP FRQ Pattern

Key Takeaways — Part 5

Part 6: Practice Workshop

Integration Applications

Integration Application Decision Guide

AP-Style Worked Problem

Key Takeaways — Part 6

Part 7: Final Assessment

Integration Applications

Complete Formula Reference

Top AP Mistakes

Integration Applications — Complete!

Integrating with Respect to $y$