Calculus

posted by .

Find the area of the region enclosed by
y−2x^2 ≥ 0 and |x|+|y| ≤ 1

  • Calculus -

    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 -

    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

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. linear programming

    Minimize z = 3x + 6y subject to: 4x + y ≥ 20 x + y ≤ 20 x + y ≥ 10 x ≥ 0 y ≥ 0 Graph the feasibility region. Identify all applicable corner points of the feasibility region. Find the point(s) (x,y) that …
  2. Calculus:)

    Find the area of the region enclosed by y−2x^2≥0 and |x|+|y| ≤ 1
  3. Calculus:)

    Find the area of the region enclosed by y−2x^2≥0 and |x|+|y| ≤ 1
  4. Calculus 2

    Find c>0 such that the area of the region enclosed by the parabolas y=x^2−c^2 and y=c^2−x^2 is 90.
  5. calculus

    Evaluate ∬R 1/sqrt(x^2+y^2) dx dy, where R is the region bounded by y≥0 and x^2−x+y^2≤0.
  6. CALCULUS

    Sketch the region enclosed by the given curves. y = tan 3x, y = 2 sin 3x, −π/9 ≤ x ≤ π/9 then then find the area. i can sketch but cant find correct area
  7. Algebra 2

    Can someone explain this inequality equation to me please?
  8. Algebra 2

    Jack is selling wristbands and headbands to earn money for camp. He earns $2 for each wristband and $3 for each headband. He wants to earn at least $50. He needs to sell at least 5 wristbands. Which system of inequalities is shown …
  9. Calculus 1

    Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y = x^−3, 2 ≤ x ≤ 5
  10. Calculus

    Sketch the region on paper. If it is a finite region, find its area. Round your answer to three decimal places. (If the area is not finite, enter NONE.) S = {(x,y) | x ≥ −4, 0 ≤ y ≤ e-x/2}

More Similar Questions