How to find the area of the region bounded by the curves y = x^4 and y = 8x ?

Question

Scarlett Allison · Accepted Answer

The area is #48/5# square units.

We start by finding their points of intersection . #x^4 = 8x# #x^4 - 8x = 0# #x(x^3 - 8) 0# #x = 0 or x^3 = 8# #x= 0 or x = 2# These will be our bounds of integration. We also see that on #[0, 2]#, #y= 8x# lies above #y = x^4# because at #x= 1# for instance #y = 8x = 8# while #y = x^4 =1#. Our expression for area will therefore be #A = int_0^2 8x - x^4dx# #A = [4x^2 - 1/5x^5]_0^2# #A = 4(2)^2 - 1/5(2)^5# #A = 16 - 32/5# #A = 48/5# Thus, the area between the two curves is #48/5#. Hopefully this helps!

Charles Arocha · Answer

undefined To find the area of the region bounded by the curves $y = x^4$ and $y = 8x$, you need to first determine the points of intersection of the two curves. Then, integrate the absolute difference between the two functions over the interval of intersection from the x-coordinate of the leftmost intersection point to the x-coordinate of the rightmost intersection point. This integral represents the area between the curves.

1. Find the points of intersection by setting the two functions equal to each other and solving for x.
2. Integrate the absolute difference between the functions over the interval determined by the intersection points.

The integral setup is:

$$Area = \int_{x_1}^{x_2} |(8x) - (x^4)| \, dx$$

Where $x_1$ and $x_2$ are the x-coordinates of the intersection points.