Applications of Definite Integrals

Spin a single curve all the way around the x-axis and it sweeps out a solid. Slice that solid into very thin disks, each a circle of radius f(x), so stacking their areas π[f(x)]² by integration gives the volume. Arc length uses the same idea: chop the curve into tiny straight pieces ds = √(dx²+dy²), measure each with the Pythagorean theorem, and add them up. Spin the rotation angle below to watch the solid form, and drag the upper limit t to watch the arc length pile up.

Volume of a Solid of Revolution

Rotation Angle270°

Function0

🍩 Stack the Disks

①Rotating a curve about the x-axis creates a solid

②At each x, the disk radius = f(x)

③Disk area = π[f(x)]² → integrate to get volume

Volume Formulas

Disk Method

V = π ∫_a^b [f(x)]² dx

Rotation about x-axis: integrate area of disks with radius f(x)

Washer Method

V = π ∫_a^b ([R(x)]² − [r(x)]²) dx

Region between two functions: outer radius R, inner radius r

Shell Method

V = 2π ∫_a^b x · f(x) dx

Rotation about y-axis: integrate cylindrical shells with radius x, height f(x)

💡 Disk vs Shell

①x-axis rotation + integrate over x → disk method

②y-axis rotation + integrate over x → shell method

③When both work, choose the easier one

Arc Length

Upper Limit t3

📏 Measure Along the Curve

①Cut the curve into tiny pieces and each becomes a straight segment (ds)

②ds = √(dx² + dy²) — Pythagorean theorem!

③Sum (integrate) all pieces to get arc length

Arc Length Formula

L = ∫_a^b √(1 + [f'(x)]²) dx

Integrate ds = √(1 + (dy/dx)²) dx

Area and Velocity·Distance

Area Between Two Curves

S = ∫_a^b |f(x) − g(x)| dx

Integrate the absolute value of (upper function − lower function)

Velocity and Distance

distance = ∫_a^b |v(t)| dt, displacement = ∫_a^b v(t) dt

|v| integral = total distance, v integral = displacement (signed)

🚗 Physical Meaning of Integration

①Integrate velocity v(t) → distance

②Integrate acceleration a(t) → velocity

③Integration is 'accumulation' — sum of moments gives the whole

Worked Examples

Example 1

Find the volume of the solid formed by rotating y = √x (0 ≤ x ≤ 4) about the x-axis.

Use the disk method V = π∫[f(x)]² dx, where [f(x)]² = x.

V = π ∫₀⁴ (√x)² dx = π ∫₀⁴ x dx

Evaluate the definite integral.

π · 4²2 = π · 8 = 8π

▸ 8π

A volume of revolution integrates the disk area π[f(x)]². For rotation about the x-axis, the disk method is standard.

Example 2

Find the area enclosed by y = x² and y = 2x.

Find the intersections to set the limits and the upper/lower functions.

x² = 2x ⇒ x = 0, 2, upper 2x · lower x²

Integrate (upper − lower).

∫₀² (2x − x²) dx = 4 − 83 = 43

▸ 43

Area integrates (upper − lower) between the intersection points. Check which function is on top first.

Summary

Three Key Formulas for Applications of Integrals

V = π∫[f(x)]²dx, L = ∫√(1+[f']²)dx, S = ∫|f−g|dx

Volume of revolution, arc length, area between curves

CSAT-style

A point P moves on a line with velocity v(t) = 3t² − 6t. What distance does P travel from t = 0 to t = 3?

①4

②6

③8

④10

⑤12

▸ ③ 8

Examine the sign of v(t) = 3t(t − 2): negative on (0,2), positive on (2,3).

v(t) = 3t(t − 2), position s(t) = t³ − 3t²

Total distance is ∫|v| dt, so add the absolute displacements over each subinterval.

|s(2) − s(0)| + |s(3) − s(2)| = 4 + 4 = 8

🎯 Exam Points

①Disk method: V = π∫[f(x)]²dx — basic x-axis rotation

②Washer: two functions → mind [R²−r²] difference

③Shell: 2π∫x·f(x)dx — useful for y-axis rotation

④Arc length: √(1+f'²) is key, find f' first

⑤Area: |f−g| absolute value → check intersection points where order changes

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