Question

sketch the region bounded by the graphs of the equations and find its area
X=y^2, y-x=2, y=-2, y=-3

Answer 1

$bounded \: area \: = \frac{9}{2}$

Explanation:

Based on the sketch .We are looking for a double integral solution to calculate the area bounded by the curves.

$x = - {y}^{2}$

$y = x + 2 = > x = y - 2$

The points of intersection are the solution of the equation..

The corresponding

y

-coordinates are:

$x = - 1 = y = 1$

$x = - 4 = y = - 2$

Giving the coordinates

(-1 ,1 ) and (-4, -2 )

If in the above diagram we look at an infinitesimally thin horizontal strip (in black) then the limits for

x

and

y

are:

$x \: varies \: from \: y \: - 2 \: to \: - {y}^{2}$

$y \: varies \: from \: - 2 \: to \: 1$

And so we can represent the bounded are by the following double integral:

.

. . Answer =

$\frac{9}{2}$

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