Examples of Double Iterated Integrals over Rectangular Regions (Example 1)

These kinds of double integrals have numbers as the integral boundaries for the x and y variables. Calculate the iterated integral: \(\int_1^4 {\int_0^2 {(6{x^2}y - 2x)dydx} } \)

Start off with the inner integral. Pay attention to the order of the variables, here it's dydx, so when doing the inside integral integrate with respect to the "y" variable. So we start by doing:

\[\int_0^2 {(6{x^2}y - 2x)dy} \]

\[\int_1^4 {3{x^2}{{(2)}^2} - 2x(2) - 3{x^2}{{(0)}^2} + 2x(0)} dx\]

\[\int_1^4 {(12{x^2} - 4x)} dx\]

\[4{(4)^3} - 2{(4)^2} - 4{(1)^3} + 2{(1)^2}\]

\[ = 222\]

Not so bad, but the first inner integral you do will be a bit of algebra so be wary of algebraic mistakes and be cautious of the order of the integral and check if it's dxdy or dydx.

Please see our other rectangular double integral examples below:

Double Iterated Integrals Rectangular Region (Ex2)

Alternatively, return to the rectangular double integral hub page here:

Double Iterated Integrals Rectangular Regions

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