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

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Answer 1 · 2018-03-13T16:11:16

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!