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Daily Math Minute

Unit 8: Applications of Integration

Area Between Curves & Volume

Finding area between curves and volume using the disc and washer methods.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Area Between Curves and Volume by Revolution

Definition — Area Between Curves, Disc Method, and Washer Method

The area between f(x) ≥ g(x) on [a,b] is ∫ₐᵇ[f(x)−g(x)]dx. Revolving a region around an axis produces a solid whose cross-sections are circles (disc method, V = ∫ₐᵇπ[R(x)]²dx) or, when the region doesn't touch the axis, rings (washer method, V = ∫ₐᵇπ([R(x)]²−[r(x)]²)dx).

Worked Example — Area Between Two Curves

Find the area between y = 6−x² and y = x²−2. Intersections: 6−x² = x²−2, so 8 = 2x², x = ±2. At x = 0, y = 6−x² gives 6, the top curve. Area = ∫₋₂²[8−2x²]dx = [8x−2x³/3]₋₂². At x=2: 16−16/3 = 32/3. At x=−2: −16+16/3 = −32/3. Area = 32/3−(−32/3) = 64/3.

Worked Example — The Disc Method

The region under y = x² from x = 0 to x = 2, revolved around the x-axis: V = ∫₀²π(x²)²dx = π∫₀²x⁴dx = π[x⁵/5]₀² = 32π/5.

Worked Example — The Washer Method

The region between y = x+2 and y = x² (intersecting at x = −1, 2, both curves nonnegative on this interval) is revolved around the x-axis. Outer radius R = x+2, inner radius r = x². V = π∫₋₁²[(x+2)²−x⁴]dx = π∫₋₁²[x²+4x+4−x⁴]dx = π[x³/3+2x²+4x−x⁵/5]₋₁². At x=2: 8/3+8+8−32/5 = 184/15. At x=−1: −1/3+2−4+1/5 = −32/15. V = π(184/15−(−32/15)) = π(216/15) = 72π/5.

Integral Visualizer

Integration bounds
0
2

Animate b (the play button above) to watch the shaded area — and the accumulation curve below — grow.

Approximation method
10

f(x), with the region between a and b shaded

Accumulation function A(x) = ∫ₐˣ f(t) dt — linked to the graph above; panning or zooming either moves both

What the Fundamental Theorem of Calculus tells us
  1. A(x) = ∫ from a to x of f(t) dt is the accumulation function. At x = b ≈ 2, A(b) ≈ 2.67 — the signed area under f from a ≈ 0 to b.
  2. The Fundamental Theorem of Calculus says A'(x) = f(x): differentiating the accumulation function gives the original function back.
  3. Confirmed numerically here: differentiating the accumulation curve gives A'(b) ≈ 4, matching f(b) ≈ 4.
Numerical approximation comparison
Every method's estimate of ∫ₐᵇ f(x) dx at n = 10 subdivisions, against a high-resolution reference — all should converge toward the same value.
MethodEstimate
Left Riemann sum2.28
Right Riemann sum3.08
Midpoint Riemann sum2.66
Trapezoidal Rule2.68
Simpson's Rule2.66667
Exact (high-resolution reference)2.66667
Analysis of f(x)
y-intercept
(0, 0)
x-intercepts
(0, 0)
Extrema
local min at (0, 0)
Inflection points
none found in view
Vertical asymptotes
none found in view
Horizontal asymptotes
none found
Domain
all real numbers in view
Range (estimated)
approximately [0, 100]

Tip

A region's area between curves equals the exact area under its difference function — the same integral, viewed as a single-function area problem, which is how a Cartesian-only graphing tool can still verify it.

Common Mistakes

  • Using the disc method's formula for a region that has a gap between it and the axis of revolution.

    Any gap between the revolved region and the axis produces a hole in the solid — the washer method, π(R²−r²), is needed instead of the disc method.

Key Takeaways

  • Area between curves and volume of revolution are both definite integrals of a cross-sectional quantity — height for area, π·(radius)² (or the difference of two) for volume.
  • The washer method is the disc method's extension to a region with a gap from the axis of revolution.

Summary

The final lesson of this unit extends the same accumulation idea to a length rather than an area or volume — arc length, unique to BC.

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