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Hint

1

Volume using cross-sectional area

∫ from a to b of A(x)dx

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2

Volume using disc method

∫ from a to b of π(f(x)²)dx

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3

Volume using washer method

∫ from a to b of π(f(x)²-g(x)²)dx

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4

Length of an unparameterized curve

∫ from a to b of √(1 + f’(x)^2)dx

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5

Work if force is constant

force * distance

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6

Work if force = F(x)

∫ from a to b of F(x)dx

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7

Work needed to stretch spring from x=a to x=b

∫ from a to b of F(x)dx

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8

Work needed to pump water from a to b

∫ from a to b of 62.5A(x)(l-x)dx

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9

bar x

(∫ from a to b of x(f(x)-g(x))dx) / A

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10

bar y

(∫ from a to b of 1/2(f(x)^2-g(x)^2)dx) / A

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11

Pappus’ Theorem

V = 2πbA

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12

Length of parameterized curve

∫ from t₀ to t₁ of √(x’(t)²+y’(t)^2)dt

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