Sure. To find the areas under the curves F(x) = x^4 and G(x) = x^3 from x=0 to x=1 and the area between these curves, we integrate the functions over the interval from 0 to 1.
Let's start with F(x) = x^4. The definite integral of x^4 from 0 to 1 is computed as follows:
∫ from 0 to 1 of x^4 dx = [ x^5 / 5 ] from 0 to 1 = [1^5 / 5 - 0^5 / 5] = 1/5.
Therefore, the area under the curve F(x) = x^4 from x=0 to x=1 is 1/5.
Next, let's compute the definite integral of G(x) = x^3 from 0 to 1:
∫ from 0 to 1 of x^3 dx = [x^4 / 4] from 0 to 1 = [1^4 / 4 - 0^4 / 4] = 1/4.
Therefore, the area under the curve G(x) = x^3 from x=0 to x=1 is 1/4.
The area between the functions F(x) and G(x) within the interval x=0 to x=1 is the absolute difference between their definite integrals. So, the area between the curves is |1/5 - 1/4| = |4/20 - 5/20| = 1/20.
So, the area under the curve F(x) = x^4 from x=0 to x=1 is 1/5, the area under the curve G(x) = x^3 from x=0 to x=1 is 1/4, and the area between the curves is 1/20.