CalcRef

Topics · 14 formulas

Applications of Integrals

Applications of Integrals in CalcRef: 14 free calculus formulas covering area, volume, arc length, surface area, and work. Each one is shown with its notation and the conditions it holds under, and links to a page with a worked example, so you can revise the topic or find the exact rule you need.

Area Between Curves

A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\, dx

The area between two curves f(x) and g(x) from x = a to x = b. Take the absolute value or integrate top minus bottom.

Area

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Disk Method

V=π∫ab[f(x)]2 dxV = \pi \int_a^b [f(x)]^2 \, dx

Volume of revolution about the x-axis when there is no gap between the curve and the axis. Each cross-section is a disk with radius f(x).

Volume

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Washer Method

V=π∫ab([R(x)]2−[r(x)]2)dxV = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right) dx

Volume of revolution when there is a gap between the curve and the axis. R(x) is the outer radius and r(x) is the inner radius.

Volume

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Shell Method

V=2π∫abx⋅f(x) dxV = 2\pi \int_a^b x \cdot f(x) \, dx

Volume of revolution using cylindrical shells. Useful when revolving about the y-axis or when the disk/washer method leads to difficult integrals.

Conditions: As written, the region under y = f(x) on [a, b] is revolved about the y-axis with 0 ≤ a < b, so the shell radius is x itself. For a different axis of revolution x = c, replace the radius x with |x - c|.

Volume

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Arc Length

L=∫ab1+[f′(x)]2 dxL = \int_a^b \sqrt{1 + [f'(x)]^2}\, dx

The length of a curve y = f(x) from x = a to x = b.

Arc Length & Surface Area

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Surface Area of Revolution

S=2π∫abf(x)1+[f′(x)]2 dxS = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2}\, dx

The surface area generated by revolving y = f(x) about the x-axis.

Conditions: f(x) ≥ 0 on [a, b].

Arc Length & Surface Area

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Work (Spring)

W=∫abkx dx=12k(b2−a2)W = \int_a^b kx\, dx = \frac{1}{2}k(b^2 - a^2)

Work done compressing or stretching a spring from position a to b, where k is the spring constant (Hooke's law: F = kx).

Work & Force

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Work (General)

W=∫abF(x) dxW = \int_a^b F(x)\, dx

Work done by a variable force F(x) over a displacement from a to b.

Work & Force

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Hydrostatic Force

F=∫abρg⋅d(y)⋅w(y) dyF = \int_a^b \rho g \cdot d(y) \cdot w(y)\, dy

The force of fluid pressure on a submerged vertical surface, where d(y) is the depth and w(y) is the width at depth y.

Work & Force

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Work (Pumping)

W=∫abρgA(y)(h−y) dyW = \int_a^b \rho g A(y)(h - y)\, dy

Work to pump fluid out of a tank. A(y) is the cross-sectional area at height y, h is the height to which fluid is pumped.

Work & Force

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Centroid (x-coordinate)

xˉ=1A∫abx[f(x)−g(x)] dx\bar{x} = \frac{1}{A}\int_a^b x[f(x) - g(x)]\, dx

The x-coordinate of the centroid of the region between f(x) and g(x). A is the area of the region.

Centroids & Averages

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Centroid (y-coordinate)

yˉ=1A∫ab12([f(x)]2−[g(x)]2)dx\bar{y} = \frac{1}{A}\int_a^b \frac{1}{2}\left([f(x)]^2 - [g(x)]^2\right) dx

The y-coordinate of the centroid of the region between f(x) and g(x).

Centroids & Averages

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Net Change Theorem

∫abF′(x) dx=F(b)−F(a)\int_a^b F'(x)\, dx = F(b) - F(a)

The integral of a rate of change gives the net change. This is just FTC Part 2 interpreted in context.

Centroids & Averages

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Probability (Continuous)

P(a≤X≤b)=∫abf(x) dxwhere ∫−∞∞f(x) dx=1P(a \leq X \leq b) = \int_a^b f(x)\, dx \quad \text{where } \int_{-\infty}^{\infty} f(x)\, dx = 1

For a continuous random variable with probability density function f(x), the probability that X falls between a and b is the integral of f from a to b.

Centroids & Averages

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See all 14 Applications of Integrals formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.