91Ó°ÊÓ

Find the area of the entire region bounded by the curves \(y=\frac{x^{3}}{x^{2}+1}\) and \(y=\frac{8 x}{x^{2}+1}\).

Evaluate the following integrals. $$\int \frac{d x}{2 x^{2}-12 x+36}$$

Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. $$\int_{0}^{2 \pi} \frac{d x}{(4+2 \sin x)^{2}}$$

$$\text {Evaluate the following integrals.}$$ $$\int_{0}^{\pi / 8} \sqrt{1-\cos 8 x} d x$$

The curves \(y=x e^{-a x}\) are shown in the figure for \(a=1,2,\) and 3 a. Find the area of the region bounded by \(y=x e^{-x}\) and the \(x\) -axis on the interval [0,4]. b. Find the area of the region bounded by \(y=x e^{-a x}\) and the \(x\) -axis on the interval \([0,4],\) where \(a > 0\). c. Find the area of the region bounded by \(y=x e^{-a x}\) and the \(x\) -axis on the interval \([0, b] .\) Because this area depends on \(a\) and \(b,\) we call it \(A(a, b),\) where \(a > 0\) and \(b > 0\). d. Use part (c) to show that \(A(1, \ln b)=4 A(2,(\ln b) / 2)\). e. Does this pattern continue? Is it true that \(A(1, \ln b)=\) \(a^{2} A(a,(\ln b) / a) ?\).

Find the volume of the following solids. The region bounded by \(y=\frac{1}{\sqrt{4-x^{2}}}, y=0, x=-1,\) ar \(x=1\) is revolved about the \(x\) -axis.

$$\text {Evaluate the following integrals.}$$ $$\int_{0}^{\pi / 4}(1+\cos 4 x)^{3 / 2} d x$$

Find the volume of the following solids. The region bounded by \(y=1 /(x+2), y=0, x=0,\) and \(x=3\) is revolved about the line \(x=-1\)

Consider the region \(R\) bounded by the graph of \(f(x)=\sqrt{x^{2}+1}\) on the interval [0,2] a. Find the volume of the solid formed when \(R\) is revolved about the \(x\) -axis. b. Find the volume of the solid formed when \(R\) is revolved about the \(y\) -axis.

Find the volume of the solid generated when the region bounded by \(y=\cos x\) and the \(x\) -axis on the interval \([0, \pi / 2]\) is revolved about the \(y\) -axis.