91Ó°ÊÓ

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by$$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$if the improper integral exists. Laplace Transforms are used to solve differential equations, find the Laplace Transform of the function. $$ f(t)=1 $$

In Exercises 93 and 94 , for the region bounded by the graphs of the equations, find (a) the volume of the solid formed by revolving the region about the \(x\) -axis and (b) the centroid of the region. \(y=\sin x, y=0, x=0, x=\pi\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(p(x)\) is a polynomial, then \(\lim _{x \rightarrow \infty}\left[p(x) / e^{x}\right]=0\).

Use integration by parts to verify the formula. (For Exercises \(89-92\), assume that \(n\) is a positive integer.) $$ \int e^{\overline{a x}} \sin b x d x=\frac{e^{a x}(a \sin b x-b \cos b x)}{a^{2}+b^{2}}+C $$

Volume The region bounded by \(y=e^{-x^{2}}, y=0, x=0\), and \(x=b(b>0)\) is revolved about the \(y\) -axis. (a) Find the volume of the solid generated if \(b=1\). (b) Find \(b\) such that the volume of the generated solid is \(\frac{4}{3}\) cubic units.

For the region bounded by the graphs of the equations, find (a) the volume of the solid formed by revolving the region about the \(x\) -axis and (b) the centroid of the region. \(y=\cos x, y=0, x=0, x=\pi / 2\)

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by$$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$if the improper integral exists. Laplace Transforms are used to solve differential equations, find the Laplace Transform of the function. $$ f(t)=t $$

Use integration by parts to verify the formula. (For Exercises \(89-92\), assume that \(n\) is a positive integer.) $$ \int e^{a x} \cos b x d x=\frac{e^{a x}(a \cos b x+b \sin b x)}{a^{2}+b^{2}}+C $$

Surface Area Find the area of the surface formed by revolving the graph of \(y=2 \sqrt{x}\) on the interval \([0,9]\) about the \(x\) -axis.

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by$$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$if the improper integral exists. Laplace Transforms are used to solve differential equations, find the Laplace Transform of the function. $$ f(t)=t^{2} $$