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Multiple integrals

Partial derivatives & multiple integrals

Japanese school year: University year 1

What you learn

Learn to calculate double integrals of functions of two variables over planar regions. They are widely used to determine the volume under surfaces and the area of two-dimensional shapes. A solid foundation in single-variable definite integration and iterated integrals is recommended before starting.

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Key points

This formula evaluates a double integral over a 2D region DD as an iterated integral. First integrate with respect to yy, then integrate the result with respect to xx.

∬Df(x,y) dxdy=∫ab(∫g1(x)g2(x)f(x,y) dy)dx\iint_D f(x,y)\,dxdy = \int_a^b \left( \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy \right) dx

When integrating over a rectangular region and the function factors into f(x)g(y)f(x)g(y), you can compute each single integral separately and multiply the two results.

∬[a,b]×[c,d]f(x)g(y) dxdy=∫abf(x) dx∫cdg(y) dy\iint_{[a,b]\times[c,d]} f(x)g(y)\,dxdy = \int_a^b f(x)\,dx \int_c^d g(y)\,dy

Integrating the constant function 1 over a region DD gives the area SS of that region. Use this formula to determine the area of irregular planar shapes.

∬Ddxdy=S\iint_D dxdy = S

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