Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 10 x4 + 5 dx, n = 4 2

Question

Keep getting the wrong answer

Answer 1

1. Calculate the Width of Each Subinterval: The interval [2,10][2, 10][2,10] has a total width of 10−2=810 - 2 = 810−2=8. Since n=4n = 4n=4, divide this interval into 4 subintervals, each of width:
2. Δx= 8 / 4= 2
3. Determine the Midpoints of Each Subinterval: The midpoints xi∗x_i^*xi∗​ for each subinterval are:
4. For the first subinterval [2,4], midpoint x1∗=3x_1^* = 3x1∗​=3
5. For the second subinterval [4,6], midpoint x2∗=5x_2^* = 5x2∗​=5
6. For the third subinterval [6,8], midpoint x3∗=7x_3^* = 7x3∗​=7
7. For the fourth subinterval [8,10], midpoint x4∗=9x_4^* = 9x4∗​=9
8. Evaluate the Function at Each Midpoint: Compute f(x)= x⁴+5 at each midpoint:
9. f(3)=34+5=81+5=86f(3) =86
10. f(5)=54+5=625+5=630f(5) = =630
11. f(7)=74+5=2401+5=2406f(7) = =2406
12. f(9)=94+5=6561+5=6566f(9) =6566
13. Apply the Midpoint Rule: Using the Midpoint Rule, approximate the integral by:
14. ∫210(x4+5) dx≈Δx⋅(f(x1∗)+f(x2∗)+f(x3∗)+f(x4∗)
15. Substituting the values:
16. ≈2⋅(86+630+2406+6566)
17. Calculate the sum inside the parentheses:
18. 86+630+2406+6566=968886 + 630 + 2406 + 6566 = 968886+630+2406+6566=9688
19. Then, multiply by Δx\Delta

So, the approximate value of the integral is:

∫₂¹⁰ (x⁴+5) dx≈19376.00