utf31-004
Derive the volumes for the following solids.
A right pyramid whose altitude is $h$ and whose base is a square
with sides of length $a.$
The water which is two inches deep in a hemispherical basin of radius
one foot.
A solid object with cross sections being squares of side length $ s(x)
= \sqrt {\sin x} 0\leq x \leq \pi.$
The intersection of two tubes of radius $r$ at right angles.
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utf33-001
Consider $ g(x) = u(x) v(x)$ where $ u(x)$ and $
v(x)$ are differentiable with respect to $x$.
Find $\frac{d { g(x)}}{dx}$.
Find an expression for $\int{u dv}$ using part (a). $$(Notice:
\int{u dv}=
\int{ u(x) \frac{d{ v(x)}}{dx} dx}=\int{ u(x) \frac{dv(x)}{dx} dx})$$
Use (b) to compute the following integrals.
\begin{alignat*}{3}
&i) \int{\frac{1+\sin x }{ \cos^2 x }} & &ii) \int{\cos^3 {(\frac{x }{ 2})}
\sin x } & &iii) \int{\frac{dx }{ 1+e^x}} \\
&iv) \int{x \sin x} & &v) \int{x^2 \ln x} & &vi) \int{x^3e^{2x}}
\end{alignat*}
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workshop By intensive workshop, we refer to the workshops originally used and developed at PDP at UC Berkeley . . . and ESP at UT Austin. There are a number of different and . . . 1K - last updated 2009-08-20 23:06 by EricHsu