math final help

Question

find the area between y=2x+3 and y=-x^2+10x-9

Answer 1

Greetings! Lets solve this problem shall we?!

Lets begin by noticing that we do not have the region to find the area so we must find the x values where this area is located. So, instead of graphing it on the calculator, we can find these x-values by setting the two equations equal to each other such that

2x+3 = -x²+10x-3

x²+2x+3 = 10x - 9

x²-8x+3 = -9

x²-8x-12 = 0

(x-6)(x-2) = 0

So x= 2 and 6 ...... These are our limits to integrate and find the area!

Then,

∫₂⁶ [(-x²+10x-9) - (2x+3)] dx = ∫⁶₂ (-x²+8x-12)dx = (-x³/3 + 4x² - 12x)⁶₂ = 32/3

Hope this helped!