Set up an integral in cylindrical coordinates, including bounds, but do not evaluate, to represent the volume in the bounded space between z=x2+y2\style{font-size:35px}{z=x^2+y^2} and z=x2+y2\style{font-size:35px}{z=\sqrt{x^2+y^2}}.

Set up an integral in cylindrical coordinates that represents the volume to the right of the cone y=x2+z2y=\sqrt{x^2+z^2} and left of the plane y=4y=4. Include bounds for your integral, but no need to evaluate.

Find the surface area of r→(r,θ)=\vec{r}(r,\theta)= with 0≤θ≤3π20\leq\theta\leq \frac{3\pi}{2} and 1≤r≤51\leq r\leq 5.Hint: ∫∫1 dS=∫∫r→r×r→θ dA\int\int{1\ dS}=\int\int{\left\|\vec{r}_r \times \vec{r}_{\theta}\right\|\ dA}.

Using polar coordinates, evaluate ∫R∫ln(x2+y2) dA\int_R\int{\ln{(x^2+y^2)}\ dA} where RR is the washer 4≤x2+y2≤94\leq x^2+y^2 \leq 9 in quadrants I, II, and III.

Evaluate ∫1e ∫ln(y)1e(x2)y dxdy\int_1^{e}\ \int_{\ln(y)}^{1}{\frac{e^{(x^2)}}{y}\ dxdy} by switching the order of integration.

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