#### To determine

**To find:**

The area of region bounded by given curves.

#### Answer

323

#### Explanation

**1) Concept:**

The area A of the region bounded by the curves x=fy& x=gy, and the lines y=c, y=d, where f and g are continuous and fy≥g(y) for c≤y≤d, then area is

A= ∫cdfy-gydy

**2) Given:**

The region bounded by the curves x+y=0 and x=y2+3y

**3) Calculation:**

As the given region is bounded by the curves x+y=0 and x=y2+3y,

From the graph, the right boundary curve is x+y=0 ⇒x=-y and the left boundary curve is

x=y2+3y

x=-y and x=y2+3y intersect each other when, -y=y2+3y

y2+4y=0, yy+4=0

y=0 or y=-4

Therefore, the area of the region bounded by the curves x=-y and x=y2+3y between x=-4 to x=0 is given by

A= ∫-40-y -y2+3ydx

= ∫-40-y2-4ydx

By using fundamental theorem of calculus and power rule,

A=-y33-4y22-40

=-y33-2y2-40

= -(0)33-2(0)2--(-4)33-2(-4)2

=0-643-32=32-643

= 96-643

A=323

**Conclusion:**

The area of the region bounded by the curves =-y and x=y2+3y between x=-4 to x=0 is A=323