Question

Approximate the area between the curve (y = 4x³) and the x-axis on the interval ([-2, 4]) using 2 trapezoids of equal width.

Answer 1

To approximate the area under the curve y = 4x³ between x = -2 and x = 4 using 2 trapezoids of equal width, calculate the function values at x = -2, x = 1, and x = 4, and then apply the trapezoidal rule to find the area of each trapezoid. The sum of these areas gives the total approximate area of 348 square units.

Step-by-step explanation:

To approximate the area between the curve y = 4x³ and the x-axis on the interval ([-2, 4]) using 2 trapezoids of equal width, we follow these steps:

1. Divide the interval [-2, 4] into two equal parts, which gives us the points x = -2, x = 1, and x = 4.
2. Calculate the value of the function at these points: y(-2) = 4(-2)³ = -32, y(1) = 4(1)³ = 4, and y(4) = 4(4)³ = 256.
3. Use the trapezoidal rule formula for each trapezoid:
   A1 = (b1 + b2)/2 ⋅ h, where A1 is the area of the first trapezoid with bases b1 = y(-2) and b2 = y(1), and height h = 1 - (-2) = 3.
   A2 = (b1 + b2)/2 ⋅ h, where A2 is the area of the second trapezoid with bases b1 = y(1) and b2 = y(4), and height h = 4 - 1 = 3.
4. The total approximate area is the sum of A1 and A2.

Let's calculate the areas: A1 = (-32 + 4)/2 ⋅ 3 = -42, A2 = (4 + 256)/2 ⋅ 3 = 390. Thus, the total approximate area is A1 + A2 = -42 + 390 = 348 square units.