Question

Please solve this question​

Answer 1

We're asked to change the order of integration in,

$\small\text{ \displaystyle\longrightarrow I=\int\limits_0^{\frac{a}{2}}\int\limits_{\frac{x^2}{a}}^{x-\frac{x^2}{a}}f(x,\ y)\ dy\ dx }$

Here the limits are,

• $\small0\leq x\leq\dfrac{a}{2}$

• $\small\dfrac{x^2}{a}\leq y\leq x-\dfrac{x^2}{a}$

Here,

• limit for x is in terms of constants

• limit for y is in terms of x

We need to change them such that,

• limit for x is in terms of y

• limit for y is in term of constants.

Graphical representation of the limits is depicted in fig. 1.

In this fig. the lines x = 0 and x = a/2, which depict the limit for x, pαss through the points (0, 0) and (a/2, a/4) respectively. Also these points are the minimum and maximum of the bounded region respectively.

So the limit for y in terms of constants is depicted by the lines y = 0 and y = a/4 and thus,

$\small\text{ \longrightarrow\underline{0\leq y\leq\dfrac{a}{4}} }$

Now consider,

$\small\longrightarrow y\geq\dfrac{x^2}{a}$

$\small\longrightarrow x^2-ay\leq0$

$\small\longrightarrow x\in[-\sqrt{ay},\ \sqrt{ay}]$

For $\smallx\geq0,$

$\small\longrightarrow x\leq\sqrt{ay}$

Consider,

$\small\longrightarrow y\leq x-\dfrac{x^2}{a}$

$\small\longrightarrow x^2-ax+ay\leq0$

$\small\text{ \longrightarrow x\in\left[\dfrac{a-\sqrt{a^2-4ay}}{2},\ \dfrac{a+\sqrt{a^2-4ay}}{2}\right] }$

$\small\text{ \longrightarrow x\in\left[\dfrac{a}{2}-\dfrac{\sqrt{a^2-4ay}}{2},\ \dfrac{a}{2}+\dfrac{\sqrt{a^2-4ay}}{2}\right] }$

For $\smallx\leq\dfrac{a}{2},$

$\small\text{ \longrightarrow x\geq\dfrac{a}{2}-\dfrac{\sqrt{a^2-4ay}}{2} }$

Thus the limit for x in terms of y will be,

$\small\text{ \longrightarrow\underline{\dfrac{a-\sqrt{a^2-4ay}}{2}\leq x\leq\sqrt{ay}} }$

Graphical representation of new limits is depicted in fig. 2.

Hence the order of integration is changed in such a way that,

$\small\text{ \displaystyle\longrightarrow\underline{\underline{I=\int\limits_0^{\frac{a}{4}}\int\limits_{\frac{a-\sqrt{a^2-4ay}}{2}}^{\sqrt{ay}}f(x,\ y)\ dx\ dy}} }$