Find the area between the curves y = x² and y=-2x²-2x. Round to 2 decimal places.

To find the area between the curves y = x² and y = -2x² - 2x, we first need to find the points of intersection. We do this by setting the two equations equal to each other and solving for x:

x² = -2x² - 2x

Combine like terms:

3x² + 2x = 0

Factor out an x:

x(3x + 2) = 0

Setting each factor equal to zero gives the solutions x = 0 and x = -2/3. These are the points where the two curves intersect.

Next, we set up the integral to find the area between the curves. We subtract the lower function from the higher function:

∫ from -2/3 to 0 (x² - (-2x² - 2x)) dx

Simplify the integrand:

∫ from -2/3 to 0 (3x² + 2x) dx

Now we can integrate:

[ x³ + x² ] from -2/3 to 0

Substitute the limits of integration:

(0³ + 0²) - ((-2/3)³ + (-2/3)²)

Simplify:

0 - ((-8/27) + 4/9)

Simplify further:

0 - (-8/27 + 12/27)

0 - 4/27

The negative sign changes the subtraction to addition:

4/27

So, the area between the curves y = x² and y = -2x² - 2x from x = -2/3 to x = 0 is approximately 0.15 (rounded to two decimal places).