Applications of Integrals formula sheet

14 rows · CalcRef · sample of the first 6 rows

Applications of Integrals
Formula	Statement	Conditions
Area Between Curves	$A = \int_a^b \|f(x) - g(x)\|\, dx$	no restriction stated
Disk Method	$V = \pi \int_a^b [f(x)]^2 \, dx$	no restriction stated
Washer Method	$V = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right) dx$	no restriction stated
Shell Method	$V = 2\pi \int_a^b x \cdot f(x) \, dx$	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\|.
Arc Length	$L = \int_a^b \sqrt{1 + [f'(x)]^2}\, dx$	no restriction stated
Surface Area of Revolution	$S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2}\, dx$	f(x) ≥ 0 on [a, b].

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