HELPPP! Use the mid-point rule with n = 2 to approximate the area of the region bounded by y equals the cube root of the quantity 16 minus x cubed y = x, and x = 0.
Please explain the answer

The approximate area of the region using the midpoint rule with n = 2 is about 2.8418 square units.

Step-by-step explanation:

We want to use the midpoint rule with n = 2 to approximate the area of the region bounded by the graphs of:

This is shown in the graph below. Our area of interest is between the green, red, and blue lines.

First, find the intersection point of the two graphs:

Therefore, we will want to approximate from x = 0 to x = 2.

Note that the red graph (cubic root) is greater than the linear function for all x in the interval [0, 2].

Since the cubic function is greater than the linear function for all x in [0, 2], the area of the region will be the linear function subtracted from the cubic function. Hence:

We will let:

The midpoint rule is given by:

Since the area is on the interval [0, 2] and n = 2, find the width:

Hence, our x-coordinates are x₀ = 0, x₁ = 1, and x₂ = 2.

Therefore, by the midpoint rule:

Substitute:

Evaluate:

Evaluate:

Hence, the approximate area of the region is about 2.8418 square units.

Notes:

The exact area is 2.80436... square units.

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