Can anyone please help me? Evaluate the definite integral. (Assume a > 0.)

Question

Here is the problem: evaluate the integral of x^3*sqrt(a^2-x^4), from 0 to a^1/2. Where do I even start?

Answer 1

The integral will be:

∫^√a₀ x³•(a²-x⁴)^1/2 dx

In this case, treat a as constant and use the u substitution:

u = a²-x⁴

du = -4x³ dx

(-1/4) du = x³ dx

For the interval 0 ≤ x ≤ √a

u = a²-(√a)⁴ = a² - a² = 0

u = a²-(0)⁴ = a²

Now we put our integral in terms of u:

(-1/4) ∫⁰_a^2 u^1/2 du = (-1/4) [(2/3) u^3/2]⁰_a^2

= (-1/4) [(2/3)(0)^3/2 - (2/3)(a²)^3/2]

= (-1/4) [ - (2/3) a³]

= (1/6) a³