91Ó°ÊÓ

The infinite region in the first quadrant between the curve \(y=e^{-x}\) and the \(x\) -axis. Find the volume of the solid generated by revolving the region about the \(y\) -axis.

Center of gravity Find the center of gravity of the region bounded by the \(x\) -axis, the curve \(y=\sec x,\) and the lines \(x=\) \(-\pi / 4, x=\pi / 4\).

Use integration by parts to obtain the formula $$ \int \sqrt{1-x^{2}} d x=\frac{1}{2} x \sqrt{1-x^{2}}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^{2}}} d x $$

The infinite region in the first quadrant between the curve \(y=e^{-x}\) and the \(x\) -axis. Find the volume of the solid generated by revolving the region about the \(x\) -axis.

Find the area of the region that lies between the curves \(y=\sec x\) and \(y=\tan x\) from \(x=0\) to \(x=\pi / 2.\)

Use the formula $$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y$$ $$y=f^{-1}(x)$$ To evaluate the integrals. Express your answers in terms of \(x\). $$\int \sin ^{-1} x d x$$

Area Find the area between the \(x\) -axis and the curve \(y=\) \(\sqrt{1+\cos 4 x}, 0 \leq x \leq \pi\).

Use the formula $$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y$$ $$y=f^{-1}(x)$$ To evaluate the integrals. Express your answers in terms of \(x\). $$\int \tan ^{-1} x d x$$

Find the centroid of the region bounded by the graphs of \(y=x+\cos x\) and \(y=0\) for \(0 \leq x \leq 2 \pi\).

Use the formula $$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y$$ $$y=f^{-1}(x)$$ To evaluate the integrals. Express your answers in terms of \(x\). $$\int \sec ^{-1} x d x$$