Question

Sketch the region R defined by 1≤ x≤ 2 and 0≤ y≤ 1/x^3
a) Find (exactly the number a such that the line x = a divides R into two parts of equal area.
b) Then find (to 3 places) the number b such that the line y = b divides R into two parts of equal area.

Answer 1

Set 2 areas equal
\[\int\limits_{1}^{a} \frac{dx}{x^3} = \int\limits_a^2 \frac{dx}{x^3}\]
integrate using power rule
\[\int\limits x^n = \frac{x^{n+1}}{n+1}\]
where n = -3
\[|_1^a \frac{-1}{2x^2} = |_a^2 \frac{-1}{2x^2} \]
apply limits
\[\frac{-1}{2a^2} - (\frac{-1}{2}) = \frac{-1}{8} - (\frac{-1}{2a^2})\]
solve for "a"
\[\frac{1}{2a^2} + \frac{1}{2a^2} = \frac{1}{2} + \frac{1}{8}\]
\[\frac{1}{a^2} = \frac{5}{8}\]
\[a = \sqrt{\frac{8}{5}} = \frac{2 \sqrt{10}}{5}\]