Trapezoidal Rule II and Simpson's Rule for Integration

To evaluate the integration of (3,-3)x⁴ dx using Trapezoidal Rule II and Simpson's Rule, the following steps can be taken:

Trapezoidal Rule II

Step 1: Determine the number of intervals (n) the range of integration must be divided into. In this case, n=2 since the range is from 3 to -3.

Step 2: Determine the width of each interval (h) by dividing the range by the number of intervals. In this case, h=(3-(-3))/2 = 3.

Step 3: Calculate the value of the function at the endpoints of each interval. In this case, f(-3)=81 and f(3)=81.

Step 4: Calculate the value of the function at the midpoint of each interval. In this case, f(0)=0.

Step 5: Use the Trapezoidal Rule II formula to calculate the approximation of the integration:

∫ (3,-3)x⁴ dx ≈ (h/2) [f(a)+2f(m)+f(b)]

= (3/2)[81+2(0)+81]

= 243

Therefore, the approximation of the integration using Trapezoidal Rule II is 243.

Simpson's Rule

Step 1: Determine the number of intervals (n) the range of integration must be divided into. In this case, n=2 since the range is from 3 to -3.

Step 2: Determine the width of each interval (h) by dividing the range by the number of intervals. In this case, h=(3-(-3))/2 = 3.

Step 3: Calculate the value of the function at the endpoints of each interval. In this case, f(-3)=81 and f(3)=81.

Step 4: Calculate the value of the function at the midpoint of each interval. In this case, f(0)=0.

Step 5: Use Simpson's Rule formula to calculate the approximation of the integration:

∫ (3,-3)x⁴ dx ≈ (h/3) [f(a)+4f(m)+f(b)]

= (3/3)[81+4(0)+81]

= 162

Therefore, the approximation of the integration using Simpson's Rule is 162.

Conclusion

In conclusion, Trapezoidal Rule II and Simpson's Rule are numerical methods that can be used to approximate the value of definite integrals. While both methods are relatively easy to use, Simpson's Rule generally provides a more accurate approximation than Trapezoidal Rule II.