If three equal subdivisions of (0,3) are used, what is the Trapezoidal Rule approximation of \int\limits^3_0 {10-x^2} \, dx 20.5 41 26 10

Question

If three equal subdivisions of (0,3) are used, what is the Trapezoidal Rule approximation of
`\int\limits^3_0 {10-x^2} \, dx`

20.5
41
26
10

Answer 1

To use the Trapezoidal Rule to approximate the area under a curve, we divide the interval into equal subdivisions, calculate the area of each trapezoid, and then sum the areas of all the trapezoids.

In this case, we are given three equal subdivisions of (0,3), so each subdivision has a width of:

Δx = (3 - 0) / 3 = 1

We are also given four possible function values, but we are not told which function these values correspond to. Therefore, we cannot calculate the exact area under the curve or the exact Trapezoidal Rule approximation.

However, we can still use the Trapezoidal Rule formula to approximate the area under the curve based on the given values. The Trapezoidal Rule formula is:

approximation = (Δx / 2) * [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + ... + f(xn)]

where Δx is the width of each subdivision, f(xi) is the function value at each endpoint of the subdivision, and n is the number of subdivisions.

For three equal subdivisions, we have n = 2. Using the given values, we get:

approximation = (1 / 2) * [20.5 + 2(41) + 2(26) + 10]

Simplifying, we get:

approximation = 123.5

Therefore, the Trapezoidal Rule approximation of the area under the curve based on the given values is approximately 123.5 square units.