Area between two curves

Q:  Find the area bound between the two curves:

X = 2y²
X + y = 1

A:  Since this are both functions of y, we will integrate with respect to y!  So, solve for X in both equations:

(1) X = 2y²
(2) X  = 1 – y

When you graph these, you will see that equation (2) is the top curve and equation (1) is the bottom curve… To find the area bound between to curves we always integrate “top curve – bottom curve”….  Our limits of integration are the points of intersection of the two curves.

How do you find where two curves intersect?  Set them equal to eachother and solve like so:

2y² = 1 – y

2y² + y – 1 = 0

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

y = 1/2 or -1

So, integrate:

∫ (1 – y) – 2y² dy (from -1 to 1/2)

∫ (1 – y) – 2y² dy = ∫ 1 – y – 2y² dy

=  y – (1/2)y² – (2/3)y³ (from -1 to 1/2)

= [ (1/2) – (1/2)(1/2)² – (2/3)(1/2)³ ] – [ (-1) – (1/2)(-1)² – (2/3)(-1)³ ]

= 1/2 – 1/8 – 1/12 + 1 + 1/2 – 2/3 = 9/8

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