What is int_0^1x^4dx?

Question

(A) Find #int_0^1x^4dx#, Draw an appropriate sketch

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Answer 1 · 2018-03-13T16:12:09

#1/5#

First, we compute the indefinite integral:

#intx^4dx#

According to the power rule, #intx^ndx=x^(n+1)/(n+1)#

Here, since #n=4#, the integral is:

#x^5/5+C#

We take this to the bounds of #[0,1]#. We have:

#[x^5/5]_0^1#

#(1^5/5)-(0^5/5)#

#1/5-0#

#=1/5#

So the area under the curve is #1/5 "units"^2#.

Sketching:
graph{x^4 [-1.208, 1.829, -0.365, 1.154]}

It makes sense.