Lösung 2.1:4e

Aus Online Mathematik Brückenkurs 2

The double inequality means that we look for the area of the region which is bounded above in the y-direction by the straight line y=x+2 and from below by the parabola y=x2.

If we sketch the line and the parabola, the region is given by the region shaded in the figure below.

As soon as we have determined the x -coordinates of the points of intersection, x=a and x=b, between the line and the parabola, we can calculate the area as the integral of the difference between the curves' y -values:

Area= bax+2−x2dx 

The curves' points of intersection are those points which lie on both curves, i.e. which satisfy both curves' equations:

y=x+2y=x2 

By eliminating y, we obtain an equation for x,

x2=x+2

If we move all x-terms to the left-hand side,

x2−x=2

and complete the square, we obtain

x−212−212=2x−212=49

Taking the root then gives that x=2123. In other words, x=−1 and x=2.

The area of the region is now given by

\displaystyle \begin{align} & \text{Area}=\int\limits_{-1}^{2}{\left( x+2-x^{2} \right)}\,dx=\left[ \frac{x^{2}}{2}+2x-\frac{x^{3}}{3} \right]_{-1}^{2} \\ & =\frac{2^{2}}{2}+2\centerdot 2-\frac{2^{3}}{3}-\left( \frac{\left( -1 \right)^{2}}{2}+2\centerdot \left( -1 \right)-\frac{\left( -1 \right)^{3}}{3} \right) \\ & =2+4-\frac{8}{3}-\frac{1}{2}+2-\frac{1}{3} \\ & =\frac{9}{2} \\ \end{align}