Question

Calculate the area of the shaded region; the area of the region above the curve y=x³ on the interval [−2,−1] and below the x-axis.. (enter the answer as a fraction)

Answer 1

To find the area of the shaded region above the curve y=x³ and below the x-axis on the interval [-2, -1], we calculate the integral of x³ from -2 to -1, which gives us -3/4, so the area is 3/4 square units after taking the absolute value.

Step-by-step explanation:

To calculate the area of the shaded region above the curve y=x³ on the interval [−2,−1] and below the x-axis, we need to integrate the function with respect to x from -2 to -1. Since the curve is below the x-axis in this interval, the integral will give us the negative of the area we are seeking. Therefore, we will take the absolute value of the integral to get the actual area.

The integral of y=x³ from -2 to -1 is:

∫_{-2}^{-1} x³ dx = [¼ x⁴]_{-2}^{-1} = (¼(-1)⁴) - (¼(-2)⁴) = (¼) - (4¼) = -¼

Since we want the positive area, the area of the shaded region is ¼ square units.