# Calculus-Area between curves

posted by Chrissy

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region.
2y=4*sqrt(x) , y=5 and 2y+4x=8

please help! i've been trying this problem the last couple days, even asked a TA for help, but i can't arrive at the right area between the curves. Thanks in advance!

1. Reiny

first of all reduce the equation to simplest form:
y = 2√x
y = 5
y + 2x = 4

Your sketch should look like a "triangle" with one side slightly curved
see:
http://www.wolframalpha.com/input/?i=plot+y%3D2*sqrt%28x%29+%2C+y%3D5+%2C+y%2B2x%3D4

we need all the intersection points,
I assume you know how to do that, ....
but I will do the hardest one:
intersect y = 2√x and y + 2x = 4
then 2√x = (4-2x)
or
√x = 2-x
by inspection, x = 1 , then in the originals , y = 2
point of intersection is (1,2)
Hoping that you get (-1/2, 5) and ( 25/4,5) for the other points

so area
= ∫(5 - (4-2x)) dx from -1/2 to 1 + ∫(5 - 2x^(1/2) dx from 1 to 25/4
= (x + x^2)| from -1/2 to 1 + (5x - 4x^(-1/2) )| from 1 to 25/4
= 1 + 1 - (-1/2 + 1/4) + (125/4 - 8/5 - (5-4) )
= 2 + 1/4 + 125/4 - 8/5 - 1
= 309/10

you better check my arithmetic on that one, should have written it out on paper first.

2. Chrissy

Thanks I figured it out my answer was 9, and it was correct

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