#### To determine

**To evaluate:**

∫01x1-x4dx by making substitution and interpret the resulting integral in terms of an area.

#### Answer

π8

#### Explanation

**1) Concept:**

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)

**2) Given:**

∫01x1-x4dx

3) **Calculation:**

We are given that

∫01x1-x4dx

Here, use the substitution method.

Substitute x2=u.

Differentiating with respect to x

2xdx=du

xdx=du2

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

=12∫011-u2 du

=12A

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

Therefore,

12∫011-u2 du=12·π4

**Conclusion:**

Therefore,

∫01x1-x4dx=π8