Final answer:
To evaluate the double integral of 5x^3cos(y^3), we establish the integration limits for x from 0 to 2√2 and for y from x^2/4 to 2, then perform the integration step-by-step.
Step-by-step explanation:
To evaluate the double integral 5x^3cos(y^3) over the region bounded by y = 2, y = x^2/4, and the y-axis, we first need to set up the integral limits. The lower bound for y is given by the parabola y = x^2/4, and the upper bound is the line y = 2. The region is also bounded by the y-axis, which means x ranges from 0 to the positive value where the parabola intersects with y = 2.
To find this intersection, we set y = 2 and solve for x in the equation y = x^2/4:
2 = x^2/4
x^2 = 8
x = ±2√2 (We only consider the positive root since x is non-negative)
Thus, the limits for x are from 0 to 2√2. The limits for y are from x^2/4 to 2. We can now set up the integral as follows:
Integrate 5x^3cos(y^3) with respect to y from x^2/4 to 2.
Integrate the result with respect to x from 0 to 2√2.
Performing these integrations will give us the value of the double integral over the specified region.