Answer
$ \frac{55}{12} $
Work Step by Step
Step 1 of 3:
Area:
Find the area bounded by the curves $ =\sqrt[]{x} $ , $ y=-\sqrt[3]{x} $
,$ y=x-2 $
Step 2 of 3:
Find the point of intersections of two curves.
$ \sqrt[]{x}=x-2 \rightarrow x=4 $
$ -\sqrt[3]{x}=x-2 \rightarrow x=1 $
Thus the curves intersect at $ =1,x=4 \rightarrow y=-1,y=2 $
.
Step 3 of 3:
Area of the shaded region is $ \int _{-1}^{0} \left( y+2+y^{3} \right) dy+ \int _{0}^{2} \left( y+2-y^{2} \right) dy $
.
$ \int _{-1}^{0} \left( y+2+y^{3} \right) dy+ \int _{0}^{2} \left( y+2-y^{2} \right) dy= \left[ \frac{y^{2}}{2}+2y+\frac{y^{4}}{4} \right] _{-1}^{0}+ \left[ \frac{y^{2}}{2}+2y-\frac{y^{3}}{3} \right] _{0}^{2} $
$ =-\frac{1}{2}+2-\frac{1}{4}+\frac{2^{2}}{2}+4-\frac{2^{3}}{3} $
$ =\frac{55}{12} $
Hence, the required area is $ \frac{55}{12} $
