Area Between Curves, Volume of Revolution and Arc Length Calculator

Integrals applied to areas, volumes and lengths.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose what you want to find and type f(x), and g(x) if you need a second curve.

Enter the lower and upper limits.

The calculator integrates numerically and gives the answer with the formula used.

Formulas used

  • A = \int_a^b|f - g|\,dx
  • V = \pi\int_a^b(f^2 - g^2)\,dx
  • s = \int_a^b\sqrt{1 + f'^2}\,dx
  • f_{avg} = \frac{1}{b - a}\int_a^bf\,dx

Worked example

With the default numbers in the calculator above:

  1. Area between the curves (always positive: the curves swap over where they cross): 9 square units
  2. Signed integral of f minus g (area counted negative where g is above f): 9
  3. The functions and limits used: f(x) = x^2, g(x) = 0, from x = 0 to 3

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want? Volume of revolution about the x axis.

  1. Volume when the region is spun about the x axis (disc or washer method): 152.6814 cubic units
  2. The same as a multiple of pi: 48.6 pi
  3. The functions and limits used: f(x) = x^2, g(x) = 0, from x = 0 to 3

Common questions

How do I find the area between two curves?

Integrate the top curve minus the bottom curve between the limits. Where they cross, split the integral.

What is the disc method?

Spin a region about an axis and add up thin discs. Each disc has volume pi times its radius squared times its thickness.

How do I find the length of a curve?

Integrate the square root of one plus the square of the derivative.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any first-year calculus text. Integrals by Simpson's rule, accurate for smooth functions.

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