∫01x1-x4dx by making substitution and interpret the resulting integral in terms of an area.
i. The substitution rule:
If u=g(x) is a differentiable function whose range is I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du. Here,g(x) is substituted as u and then g’(x)dx =du
ii. Indefinite integral
∫xn dx=xn+1n+1+C (n≠-1)
We are given that
Here, use the substitution method.
Differentiating with respect to x
The limits change and the new limits of integration are calculated by substituting
for x=0, u=02=0& for x=1, u=12 =1
Therefore, the given integral becomes
Area A is the shaded region in the graph. It is the area of a quarter of a circle with radius 1
So, the required area is 14th area of circle of radius 1. So A=π4