Find the area of the region enclosed by the curves y=3x^2, y=8x^2, 4x+y=4 in the first quadrant

Question

Help, please, any work would be nice. Thank you in advance!:)

Answer 1

Strategy:

 

Divide the region into two pieces.

 

Find the the points where y= 8x² and y= 3x² each intersect the line. Graph the three functions to see the region of interest and use your Algebra II skills to find the intersection pts: @ x = 1/2 and x = 2/3.

 

Use the FTC to integrate between the curves for the left piece x ε [0,1/2] and integrate between the line and 3x² for the right piece x ε [1/2,2/3] and add the two values together.

 

R₁ + R₂ =  ∫_[0,1/2](8x²-3x²)dx + ∫_[1/2,2/3](-4x+4-3x²)dx

 

and R₁ + R₂ = 5/24 - 23/216 = 11/108