Question

Sketch the region of integration and change the order of integration. $\int_{0}^{{\color{red}16}} \int_{0}^{\sqrt{x}} f(x, y)dy dx$

Answer 1

The integral after the change of order of integration is $\int\limits^4_0 {\int\limits^{y^2}_0 {f(x,y)} \, dx } \, dy$.

What is meant by changing the order of integration?

• The order of integration is responsible for the description of the region and accordingly the limits of the integration.
• Here the change of order of integration implies the change of limits of integration.
• If the region of integration consisted of a vertical strip and slides along the x-axis then in the changed order a horizontal strip and a slide along the y-axis are to be considered and vice-versa.

Calculation:

Given integration is

$\int\limits^{16}_0 {\int\limits^{\sqrt{x}}_0 {f(x,y)} \, dy } \, dx$

Drawing the region of the given integration by taking the curves from the limits as y = √x ⇒ x = y² and y = 0; and the lines x = 0 and x = 16

The region of the given integration is shaded in red color.

Then change the order of the integration by fixing the axes,

as y = 0 then x = 0, and y = 4 then x = y²

Thus, the changed limits are:

x: 0 to y²

y: 0 to 4

Then the new integration with the changed order of integration is

$\int\limits^4_0 {\int\limits^{y^2}_0 {f(x,y)} \, dx } \, dy$

Learn more about changing the order of integration here:

brainly.com/question/14529241

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