Question

Find the area under the curve y = 25/x3 from x = 1 to x = t. Evaluate the area under the curve for t = 10, t = 100, and t = 1000. t = 10 t = 100 t = 1000 Find the total area under this curve for x ≥ 1.

Answer 1

Answer:

12.375 for t=10, 12.49875 for t=100, 12.49999 for t=1000, 12.5 for $x\geq 1$.

Step-by-step explanation:

One way to find the area under the curve $y = 25/x^3$ from x = 1 to x = t, is to calculate the next integral $\int_{1}^{t}\frac{25}{x^3} dx = \int_{1}^{t} 25x^{-3} dx = 25\frac{x^{-2}}{-2}|_{1}^{t} = (-25/2)x^{-2}|_{1}^{t} = (-25/2)[t^{-2}-1]$.

For t = 10 we have $(-25/2)[\frac{1}{10^2}-1] = 12.375$.

For t = 100 we have $(-25/2)[\frac{1}{100^2}-1] = 12.49875$.

For t = 1000 we have $(-25/2)[\frac{1}{1000^2}-1] = 12.49999$.

The total area under this curve for $x\geq1$ is given by

$lim_{t\to\infty}(-25/2)[t^{-2}-1] = (-25/2)(-1) = 25/2 = 12.5$