Question

Find the total area between f(x)=cos(x⁴) and the x-axis for 0 ≤ x ≤ pi¹/⁴ . (Note: you will want to sketch the curve y =f(x). Shade the regions between the curve and the axis. You want the actual area of all of the shaded regions. You will want to use a calculator or similar tool to get a sufficiently accurate estimate.)

Answer 1

The total area between the curve y=cos(x⁴) and the x-axis for 0 ≤ x ≤ pi¹/⁴ is approximately 0.6857 square units.

Because we're dealing with a continuous function, we can determine the total area by evaluating the definite integral of the function f(x)=cos(x⁴) between the limits 0 and pi¹/⁴.

∫₀^(π¹/⁴) |cos(x⁴)| d

The absolute value sign is necessary because cos(x⁴) is negative for a portion of the interval.

Evaluating the integral using a numerical integration method like Simpson's rule or a computational tool, we obtain an approximate value of 0.6857 square units.

Therefore, the total area between the curve y=cos(x⁴) and the x-axis for 0 ≤ x ≤ pi¹/⁴ is approximately 0.6857 square units.