Question

Estimate the integral ∫10sin²(π/3x)dx by the trapezoidal rule using n = 4.

Answer 1

To estimate the integral using the trapezoidal rule, divide the interval into subintervals, evaluate the function at their endpoints, and apply the formula. Plug in the provided values to find the estimated integral.

Step-by-step explanation:

To estimate the integral ∫10sin²(π/3x)dx using the trapezoidal rule, we first divide the interval [0, 10] into four subintervals. The width of each subinterval is Δx = (b-a)/n = (10 - 0)/4 = 2.

Next, we evaluate the function at each subinterval's endpoints and calculate the sum of the function values multiplied by the width of each subinterval.

Using the trapezoidal rule formula: ∫f(x)dx ≈ Δx/2 * [f(x₀) + 2f(x₁) + 2f(x₂) + f(x₃)]

Plugging in the values from the question, we have:

∫10sin²(π/3x)dx ≈ Δx/2 * [f(0) + 2f(2) + 2f(4) + f(6) + f(8) + f(10)]

Answer 2

To estimate the integral ∫10sin²(π/3x)dx by the trapezoidal rule using n = 4, divide the interval [0, 10] into four subintervals of equal width and evaluate the function at the endpoints of each subinterval. Use the trapezoidal rule formula to estimate the integral.

Step-by-step explanation:

To estimate the integral ∫10sin²(π/3x)dx by the trapezoidal rule using n = 4, we first need to divide the interval [0, 10] into four subintervals of equal width. In this case, each subinterval would have a width of (10 - 0)/4 = 2 units.

Next, we evaluate the function at the endpoints of each subinterval: x = 0, x = 2, x = 4, x = 6, and x = 8. In this case, the function values would be sin²(π/3*0) = 0, sin²(π/3*2) = 1/4, sin²(π/3*4) = 1, sin²(π/3*6) = 1/4, and sin²(π/3*8) = 0, respectively.

Finally, we use the trapezoidal rule formula to estimate the integral: ((2/2)(0 + 1/4 + 1 + 1/4 + 0)) = (1/2)(5/2) = 5/4 = 1.25 units.