∮_0^1▒∫_1^2▒〖(x^2+y^2)dxdy〗using simpson's one third rule
To evaluate the double integral ∮_0^1∫_1^2 (x^2+y^2)dxdy using Simpson's 1/3 rule, we need to divide the region of integration into subintervals and use the formula for Simpson's 1/3 rule to approximate the integral in each subinterval.
Here's how you can approach it:
Step 1: Divide the region of integration [0, 1] × [1, 2] into smaller subintervals.
- Choose an even number of subintervals in both x and y directions to apply Simpson's 1/3 rule.
- Let's say we choose 4 subintervals in each direction, resulting in a total of 16 smaller rectangles.
Step 2: Calculate the step size (h) for each direction.
- For the x-direction, h_x = (1 - 0) / 4 = 0.25
- For the y-direction, h_y = (2 - 1) / 4 = 0.25
Step 3: Construct a grid of points within each subinterval.
- The x-values in the x-direction will be: 0, 0.25, 0.5, 0.75, 1.0
- The y-values in the y-direction will be: 1, 1.25, 1.5, 1.75, 2.0
Step 4: Evaluate the function f(x, y) = x^2 + y^2 at each point in the grid.
- Calculate f(x, y) for each grid point. For example, at (x, y) = (0, 1), f(0, 1) = 0 + 1^2 = 1.
Step 5: Use Simpson's 1/3 rule formula to approximate the integral in each subinterval.
- Apply Simpson's 1/3 rule for each subinterval.
- The formula for Simpson's 1/3 rule is:
∫_a^b f(x)dx ≈ ((b - a) / 6) * (f(a) + 4f((a+b)/2) + f(b))
- Apply this formula to approximate the integral of (x^2 + y^2) with respect to x in each y subinterval.
Step 6: Sum up the approximated integrals from each subinterval.
- Add up the approximated integrals from each subinterval to find the overall approximation of the double integral.
By following these steps, you can evaluate the double integral ∮_0^1∫_1^2 (x^2+y^2)dxdy using Simpson's 1/3 rule.