83. What is the area of the region in the first quadrant enclosed by the graphs of y = cosx, y = x,and the y-axis?
(A) 0.127
(B) 0.385
(C) 0.400
(D) 0.600
(E) 0.947
First sketch the required area, see attachment. The required area is theOriginally Posted by frozenflames
shaded region in the figure.
Just counting squares indicates that the area is close to 0.4, but we
will find this area by integration. This is:
$\displaystyle
\int_0^{x_1} \cos(x)-x\ dx
$
where $\displaystyle x_1$ is the solution of $\displaystyle \cos(x)=x$ in $\displaystyle [0,1]$.
Now using the bisection method gives that $\displaystyle x_1\approx 0.739
$
So we seek:
$\displaystyle
\int_0^{0.739} \cos(x)-x\ dx=\left[\sin(x)-x^2/2\right]_0^{0.739}=0.400
$
RonL