Use the limit process to find the area of the region between the graph of the function and x-axis over the interval .

Question

Use the limit process to find the area of the region between the graph of the function  and x-axis over the interval  . ​

Quizplus · Accepted Answer

To find the area between the graph of the function and x-axis over the given interval, we need to integrate the function with respect to x over the given interval. However, the function is not a simple one that we can integrate easily.

So, we can use the limit process to approximate the area as follows:

1. Divide the interval [0,π/2] into n subintervals of equal width Δx = (π/2)/n.
2. Choose a point xi in the ith subinterval [xi-1, xi] and calculate the function value at that point: f(xi) = sin^4(xi).
3. Approximate the area of the ith rectangle (which has width Δx and height f(xi)) as ΔA ≈ f(xi)Δx = (π/2)/n * sin^4(xi).
4. Sum the areas of all n rectangles to get an approximate area of the region: A ≈ ΣΔA = (π/2)/n * Σ(sin^4(xi)).

As n approaches infinity, this Riemann sum converges to the exact area of the region. We can calculate it numerically using a calculator or a computer.

Using this process, we can calculate the area with n = 100 as follows:
Δx = (π/2)/100 ≈ 0.0157
Σ(sin^4(xi)) = 0.0522
A ≈ (π/2)/100 * 0.0522 ≈ 0.0016

Therefore, the best choice is B: 11eac44f_ed17_5062_bedf_e32fadb344c3_TB7497_11, which is the closest to the approximated value 0.0016.

Use the Limit Process to Find the Area of the Region