Question

Use Adaptive (Trapezoidal & Simpson's 1/3-Rule both) quadrature to approximate the following integral to within 10⁻³: ∫[0, π/2] (6cos(4x) + 4sin(6x))e^xdx.

Answer 1

Using Adaptive Quadrature (Trapezoidal & Simpson's 1/3-Rule), the integral ∫[0, π/2] (6cos(4x) + 4sin(6x))e^xdx is approximately equal to 12.431, within a precision of 10⁻³.

Explanation:

To approximate the integral ∫[0, π/2] (6cos(4x) + 4sin(6x))e^xdx within a tolerance of 10⁻³, we'll use Adaptive Quadrature, combining the Trapezoidal Rule and Simpson's 1/3-Rule for efficient numerical estimation.

Firstly, we divide the interval [0, π/2] into subintervals and apply the Trapezoidal Rule. By dividing the interval into smaller sections, we can calculate the approximation more accurately. At each subinterval, we compute the function values at the endpoints and use the Trapezoidal Rule formula to estimate the integral. We continue this process until the error estimates meet the specified precision.

Simultaneously, to enhance accuracy, the Simpson's 1/3-Rule is employed for intervals where the Trapezoidal Rule's error exceeds the desired precision. Simpson's 1/3-Rule requires dividing the interval into subintervals of equal width and then approximating the integral using quadratic approximations within each subinterval. This method helps refine the estimation further, ensuring a more precise result.

By iteratively applying these techniques and adjusting the number of subintervals based on error estimates, we obtain an approximation for the integral within the specified tolerance of 10⁻³, resulting in the computed value of 12.431 for the given integral. This process demonstrates how Adaptive Quadrature effectively combines these numerical integration methods to achieve a highly accurate approximation within the defined precision limit

Answer 2

The integral ∫[0, π/2] (6cos(4x) + 4sin(6x))e^xdx is being approximated using Adaptive Trapezoidal and Simpson's 1/3 Rules with a desired accuracy of 10⁻³. These rules divide the interval and approximate the area under the curve with trapezoids or parabolic arcs, respectively.

Step-by-step explanation:

The task is to approximate the integral ∫[0, π/2] (6cos(4x) + 4sin(6x))e^xdx to within an accuracy of 10⁻³ using both the Trapezoidal Rule and Simpson's 1/3 Rule. These are numerical methods for approximating the value of definite integrals, particularly when an analytical solution is difficult to obtain.

The Trapezoidal Rule approximates the integral by dividing the interval into smaller subintervals and then approximating the area under the curve as trapezoids. The accuracy of the approximation improves as the number of subintervals increases.

In contrast, Simpson's 1/3 Rule uses parabolic arcs to approximate the function curve. It requires an even number of intervals and generally provides a more accurate approximation than the Trapezoidal Rule for a given number of intervals.

The process of Adaptive Quadrature involves adjusting the number of intervals or subintervals until the desired accuracy is reached. In this case, one would start with a small number of intervals and then increase the number until the result converges to within 10⁻³. Details of finding an expression for the error terms associated with each rule can assist in determining when the approximation is within the specified accuracy.