Question

Find the area under the curve y = 21/x³ from x = 1 to x = t. Evaluate the area under the curve for t = 10, t = 100, and t = 1000.

1) t = 10
2) t = 100
3) t = 1000

Answer 1

To find the area under the curve y = 21/x³ from x = 1 to x = t, we need to integrate the function from 1 to t. The area under the curve for t = 10, t = 100, and t = 1000 is 6.3 units, 6.069 units, and 6.007 units, respectively.

Step-by-step explanation:

To find the area under the curve y = 21/x³ from x = 1 to x = t, we need to integrate the function from 1 to t. The integral of y = 21/x³ is given by A = ∫(21/x³) dx = -7/x² + C. Evaluating this expression between 1 and t, we have A = [-7/t² + C] - [-7/1² + C] = -7/t² + 7.

For t = 10, the area under the curve is A = -7/10² + 7 = 6.3 units.

For t = 100, the area under the curve is A = -7/100² + 7 = 6.069 units.

For t = 1000, the area under the curve is A = -7/1000² + 7 = 6.007 units.