91Ó°ÊÓ

Evaluate the integral and interpret it as the area of a region. Sketch the region. $$ \int_{0}^{\pi / 2}|\sin x-\cos 2 x| d x $$

Use a graph to find approximate \(x\) -coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the \(x\) -axis the region bounded by these curves. $$ y=1+x e^{-x^{3}}, \quad y=\arctan x^{2} $$

Evaluate the integral and interpret it as the area of a region. Sketch the region. $$ \int_{-1}^{1}\left|3^{x}-2^{x}\right| d x $$

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $$ y=x^{3} \sin x, y=0,0 \leqslant x \leqslant \pi ; \text { about } x=-1 $$

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \(y=-x^{2}+6 x-8, y=0 ;\) about the \(y\) -axis

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $$ y=\sin ^{2} x, y=0,0 \leqslant x \leqslant \pi ; \quad \text { about } y=-1 $$

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $$ y=x, y=x e^{1-x / 2} ; \quad \text { about } y=3 $$

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \(y=-x^{2}+6 x-8, y=0 ;\) about the \(x\) -axis

Use a graph to find approximate \(x\) -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. $$ y=3 x^{2}-2 x, \quad y=x^{3}-3 x+4 $$

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. \(y^{2}-x^{2}=1, y=2 ;\) about the \(x\) -axis