Final answer:
To find the area under the curve y = 15/x³ from x = 1 to x = t, we can use integration. The area under the curve for t = 10 is 4.95, for t = 100 is 4.9995, and for t = 1000 is 4.999995. The total area under the curve for x ≥ 1 is 5.
Step-by-step explanation:
To find the area under the curve y = 15/x³ from x = 1 to x = t, we can use integration. The integral of 15/x³ with respect to x is -5/x².
To evaluate the area under the curve for t = 10, we subtract the value of the integral at x = 1 from the value of the integral at x = 10. This gives us (-5/10²) - (-5/1²) = -0.05 - (-5) = -0.05 + 5 = 4.95. Similarly, for t = 100, we have (-5/100²) - (-5/1²) = -0.0005 + 5 = 4.9995. Finally, for t = 1000, we have (-5/1000²) - (-5/1²) = -5e-6 + 5 = 4.999995.
To find the area under the curve y = 15/x³ from x = 1 to x = t, we use integration. The area A can be found by integrating the function with respect to x:
A = ∫ y dx = ∫ 15/x³ dx
Upon integration, we get:
A = - ½ * 15/x² | from 1 to t
which simplifies to:
A = -7.5/t² + 7.5
Evaluating the area under this curve for t = 10, t = 100, and t = 1000, we get:
For t = 10, A = -7.5/10² + 7.5 = 7.425
For t = 100, A = -7.5/100² + 7.5 = 7.4925
For t = 1000, A = -7.5/1000² + 7.5 = 7.49925
The total area under the curve for x ≥ 1 would theoretically approach 7.5 as t approaches infinity.