Posted by **Lindsay** on Friday, August 12, 2011 at 4:56am.

Find the area of the region enclosed by

y−2x^2 ≥ 0 and |x|+|y| ≤ 1

- Calculus -
**Reiny**, Friday, August 12, 2011 at 7:38am
the graph of |x|+|y| ≤ 1 is a square with vertices

(1,0), (0,1), (-1,0) and (0,-1)

y ≥ 2x^2 is the region above y = 2x^2

so we need their intersection:

The line of the square in the first quadrant is

y = -x + 1

then 2x^2 = -x + 1

2x^2 + x - 1 = 0

(2x-1)(x+1) = 0

x = 1/2 or x = -1

from the diagram we want x=1/2, then y = 1/2)

From symmetry we can take the region form x = 0 to x = 1/2, then double that answer.

Area = ∫(-x + 1 - 2x^2) dx from 0 to 1/2

= -x^2/2 + x - 2x^3/3 | from 0 to 1/2

= -1/8 + 1/2 - 1/12 - 0

= 7/24

- continuation - Calculus -
**Reiny**, Friday, August 12, 2011 at 7:39am
oops, forgot to double our answer ...

since this was only the area in the first quadrant, the area in II would be the same, so the

total area = 14/24 or

7/12 square units

## Answer this Question

## Related Questions

- Calculus:) - Find the area of the region enclosed by y−2x^2≥0 and |x...
- Calculus:) - Find the area of the region enclosed by y−2x^2≥0 and |x...
- CALCULUS - Sketch the region enclosed by the given curves. y = tan 3x, y = 2 sin...
- linear programming - Minimize z = 3x + 6y subject to: 4x + y ≥ 20 x + y...
- calculus - Evaluate ∬R 1/sqrt(x^2+y^2) dx dy, where R is the region ...
- Calculus 2 - Find c>0 such that the area of the region enclosed by the ...
- calculus HELP - Find the values of a and b that make f continuous everywhere. f(...
- algebra - Identify the vertices of the solution region of the system of linear ...
- math - Maximize and minimize p = x + 2y subject to x + y ≥ 8 x + y ≤...
- math - Maximize z = 16x + 8y subject to: 2x + y ≤ 30 x + 2y ≤ 24 x...