please help area

Question

Find the area of the region between two curves below.

 

y=2x^2

 

y=x^2+4

Answer 1

Assuming the two functions are y=2x² and y = x² + 4.

 

It is necessary to find the x-coordintates of the points of intersection so that we know the interval over which to integrate. It is also helpful to sketch the graph.

 

Substitute 2x² into the 2nd function for y and solve for x. This gives +/- 2 for x.

 

Check out: https://www.desmos.com/calculator/1qxuyya9mm

 

The 2nd function is actually "above" the first one.

 

So (hard to display limits of integration as -2 to 2--at least so far I have not figured it out), so the following is sort of messy:

 

∫² [(x²+4) - 2x²]dx = ∫ (4-x²) = 4x - x³/3 ]_-2² = [(8 - 8/3) - (-8 + 8/3)] = 16 - 16/3 = 48/3 - 16/3 = 32/3.

-2