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svet-max [94.6K]
2 years ago
15

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$
Mathematics
1 answer:
skelet666 [1.2K]2 years ago
3 0

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

<h3>What is meant by changing the order of integration?</h3>
  • 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.

<h3>Calculation:</h3>

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

#SPJ4

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