Chapter 5, Problem 16PS

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

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

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

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

(c) Use tlie 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

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