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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

yy=x+2=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−212−212x−212=2=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

Area=2−1x+2−x2dx= 2x2+2x−3x3 2−1=222+22−323−2(−1)2+2(−1)−3(−1)3=2+4−38−21+2−31=29.