Get the solution to "find the area of the region that is..."

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2022-05-05

find the area of the region that is bounded by the graphs of x = y^2 and x = 2 − y

user_27qwe

Skilled2022-07-02Added 375 answers

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:

$x = - {(x + 2)}^{2}$

$∴ x = - (x^{2} + 4 x + 4)$

$∴ x^{2} + 5 x + 4 = 0$

$(x + 1) (x + 4) = 0$

$x = - 1, - 4$

The corresponding y-coordinates are:

$x = - 1 \Rightarrow y = 1$

$x = - 4 \Rightarrow y = - 2$

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:

$A = \int \int_{R} d A$
$= \int_{- 2}^{1} \int_{y - 2}^{- y^{2}} d x d y$

We can calculate the inner integral:

$\int_{y - 2}^{- y^{2}} d x = {[x]}_{y - 2}$

$= (- y^{2}) - (y - 2)$
$= - y^{2} - y + 2$
$= - (y^{2} + y - 2)$

And so:

$= \int_{- 2}^{1} - (y^{2} + y - 2) d y$

$= \frac{9}{2}$

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