Calculus help :)

Question

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

y = x⁻³,    2 ≤ x ≤ 6

Answer 1

use an online graphing calculator, like Desmos

it's a rectangular hyperbola shape, with x and y axes as asymptotes

between x=2 and x=6, the curve is very close to the x axis, leaving a small fraction as the area below the curve

y = x^-3

increase the exponent by 1, and divide by that same number. -3+1 =-2

integral = (x^-2)/(-2) = (-1/2)(x^-2) = (-1/2)(1/x^2)

evaluated between 2 and 6

=(-1/2)(1/6^2 - (1/2^2)

= (-1/2)(1/36 - 1/4)

= (1/2)(1/4 - 1/36)

= (1/2)(9/36 - 1/36)

= (1/2)(8/36)

= 4/36

= 1/9 = the exact area under the curve between x=2 and 6

which looks about right compared to relatively small fraction the graph of the curve shows