Expert: Paul Klarreich - 11/26/2007

Question
Find the area of the region bounded by the curves:
y = 8x^2
y = the square root of x

Answer
Questioner:   Alex
Category:  Calculus
Private:  No
 
Subject:  calculus: the definite integral
Question:  Find the area of the region bounded by the curves:
y = 8x^2
y = the square root of x
....................................
Hi, Alex,

Find the intersections.   Set

8x^2 = sqrt(x)   and solve.

Square both sides:

64x^4 = x

64x^4 - x = 0

x(64x^3 - 1) = 0

x = 0  and  x = 1/4 are the solutions, and if you look at the graph, you see that sqrt(x) is higher in [0,1/4] and the area is small.  So you want:

{1/4
|    [sqrt(x) - 8x^2] dx
}0

{1/4
|    [x^1/2 - 8x^2] dx
}0

x^3/2    8x^3
------ - -----
3/2        3


2x^3/2    8x^3
------ - -----, from  0 to 1/4
 3        3

2(1/4)^3/2    8(1/4)^3
----------- - ---------
 3             3

2(1/8)     8(1/64)
------- - ---------
 3           3

1/4     1/8
----- - ----
3       3

1        1
----- - ----
12      24

= 1/24

I told you it was small.