Chapter 5, Problem 9PS

Area Consider the three regions A, B, and C determined by the graph of $f(x)=\arcsin x$ as shown in the figure.

(a) Calculate the areas of regions A and R.

(b) Use your answers in part (a) to evaluate the integral

${\displaystyle {\int }_{1/2}^{\sqrt{2}/2}\arcsin xdx}$

(c) Use the methods in part (a) to evaluate the integral

${\displaystyle {\int }_1^3\ln xdx.}$

(d) Use the methods in part (a) to evaluate the integral

${\displaystyle {\int }_1^{\sqrt{3}}\arctan xdx}$