To solve the problem where we want to find the sum of the cubes of the integers from 1 to 19, we can approach it step-by-step and compute each cube before summing them up.
Let's start by calculating the cubes of each integer from 1 to 19 individually:
1. [tex]\(1^3 = 1\)[/tex]
2. [tex]\(2^3 = 8\)[/tex]
3. [tex]\(3^3 = 27\)[/tex]
4. [tex]\(4^3 = 64\)[/tex]
5. [tex]\(5^3 = 125\)[/tex]
6. [tex]\(6^3 = 216\)[/tex]
7. [tex]\(7^3 = 343\)[/tex]
8. [tex]\(8^3 = 512\)[/tex]
9. [tex]\(9^3 = 729\)[/tex]
10. [tex]\(10^3 = 1000\)[/tex]
11. [tex]\(11^3 = 1331\)[/tex]
12. [tex]\(12^3 = 1728\)[/tex]
13. [tex]\(13^3 = 2197\)[/tex]
14. [tex]\(14^3 = 2744\)[/tex]
15. [tex]\(15^3 = 3375\)[/tex]
16. [tex]\(16^3 = 4096\)[/tex]
17. [tex]\(17^3 = 4913\)[/tex]
18. [tex]\(18^3 = 5832\)[/tex]
19. [tex]\(19^3 = 6859\)[/tex]
We now list out all these cubed values:
[tex]\[1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859\][/tex]
Next, we sum these values:
[tex]\[ \begin{aligned}
1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 + 1331 + 1728 + 2197 + 2744 + 3375 + 4096 + 4913 + 5832 + 6859 &= 36100
\end{aligned} \][/tex]
Therefore, the sum of the cubes of the integers from 1 to 19 is [tex]\(36,100\)[/tex].
