91Ó°ÊÓ

Find the volume generated by rotating about the \(x\) -axis the regions bounded by the graphs of each set of equations. $$y=2 \sqrt{x}, x=1, x=2$$

Solve. $$\frac{d y}{d x}=5 x^{4} y$$

Find \(k\) such that each function is a probability density function over the given interval. Then write the probability density function. $$f(x)=k x^{2}, \quad[-2,2]$$

Find the accumulated present value of cach continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) connpounded continuously. $$R(t)=\$ 520,000, \quad T=25 y r, \quad k=6 \%$$

Determine whether each improper integral is convergent or divergent, and calculate its value if it is convergent. $$\int_{5}^{\infty} x^{4} d x$$

Find \(k\) such that each function is a probability density function over the given interval. Then write the probability density function. $$f(x)=k, \quad[1,7]$$

Determine whether each improper integral is convergent or divergent, and calculate its value if it is convergent. $$\int_{0}^{\infty} x e^{x} d x$$

Find the accumulated present value of cach continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) connpounded continuously. $$R(t)=\$ 5200 t, \quad T=18 y r, \quad k=7 \%$$

Find the volume generated by rotating about the \(x\) -axis the regions bounded by the graphs of each set of equations. $$y=\sqrt{4-x^{2}}, x=-2, x=2$$

Explain why both consumers and producers feel good when consumer and producer surpluses exist.