Question

Evaluate the definite integral `int_2^4(x-3)^3dx` ?

a) Evaluate the integral `int_2^4(x-3)^3dx`
b) Draw a graph of the function over an appropriate domain, and explain why this answer must necessarily be what it is.

Answer 1

Integral is `0`.

Explanation:

Let `x-3=t` then `dx=dt` and

`int_2^4(x-3)^3dx=int_-1^1t^3dt`

= `[t^4/4]_-1^1`

= `[1/4-1/4]=0`

Note that when `x=2`, `t=-1` and when `x=4`, `t=1`.

Now observe the graph of `(x-3)^3`, as given below

graph{(x-3)^3 [0.542, 5.542, -1.2, 1.3]}

Note that area of curve in the interval `[2,3]` is symmetric w.r.t. area of curve in the interval `[3,4]`, but they are in opposite direction, Hence the integral is `0`.