(5 points) Evaluate the integral \(\displaystyle{ \int_{1}^{2}{ \int_{0}^{\ln(x)}{ x^3 e^y ~dy } ~dx } }\).

\( 49/20 \)

(10 points) Sketch the region of integration and set up (but do not evaluate) the integral \(\displaystyle{ \iint\limits_R{ y^2 ~dA } }\) where R is the region bounded by \(y=2, y=2-x\) and \(y=x-2\).

The region is shown in the plot above. It is better to describe the region horizontally since this allows us to set up only one integral. In this case, \(0 \leq y \leq 2\) and \(2-y \leq x \leq y+2\). So our integral is
\(\displaystyle{\int_{0}^{2}{\int_{2-y}^{y+2}{ y^2 ~dx} ~dy}}\)

(10 points) Sketch the region of integration and set up (but do not evaluate) this integral in polar coordinates \(\displaystyle{ \int_{-5}^{5}{ \int_{0}^{\sqrt{25-y^2}}{ 16-x^2-y^2 ~dx }~dy } }\)

The region is shown in the plot above. From the integral, we can determine the ranges on x and y as \(-5 \leq y \leq 5\) and \(0 \leq x \leq \sqrt{25-y^2}\). From the last inequality for x, we can determine that we are working with a circle of radius 5. The range on y may give us the impression that we need the entire circle. However, since x starts at zero, this tells us that we have only the right half of the circle.
The standard equation for polar coordinates is \(r^2=x^2+y^2\). In our case, \(r=5\) and so our range on r is \(0\leq r\leq 5\). Since we want only the right half of the circle, our range on the angle is \(-\pi/2 \leq \theta \leq \pi/2\). Now we can set up our integral.
\(\displaystyle{\int_{-\pi/2}^{\pi/2}{\int_{0}^{5}{ (16-r^2)~rdr}~d\theta}}\)
Since both sets of limits of integration involve real numbers (no variables), the order of integration can be switched with no change to the integrand, i.e. \(\displaystyle{\int_{0}^{5}{\int_{-\pi/2}^{\pi/2}{(16-r^2)~rd\theta}~dr}}\) is also a correct answer.

(5 points) For the integral in the previous question in rectangular coordinates, set up (but do not evaluate) an integral (also in rectangular coordinates) by switching the order of integration.

Looking at the plot, we can see that in the previous question integration was done horizontally (since y ranges from -5 to 5). So we need to integrate vertically. This means that the range on x is \(0 \leq x \leq 5\) and y ranges from \(-\sqrt{25-x^2}\) to \(\sqrt{25-x^2}\). The integrand does not change, so our answer is \(\displaystyle{ \int_{0}^{5}{ \int_{-\sqrt{25-x^2}}^{\sqrt{25-x^2}}{ 16-x^2-y^2 ~ dy} ~dx } }\)

(10 points) Evaluate \(\displaystyle{ \int_{0}^{\pi}{ \int_{0}^{y}{ \int_{0}^{\sin(x)}{ dz } ~dx } ~dy } }\)

[ Inside Integral ]
\(\displaystyle{ \int_{0}^{\sin(x)}{ dz } = \sin(x) }\)

[ Middle Integral ]
\(\displaystyle{\int_{0}^{y}{\sin(x)~dx} = \left. -\cos(x) \right|_{0}^{y} = -\cos(y)+\cos(0) = 1-\cos(y)}\)

[ Outside Integral ]
\(\displaystyle{\int_{0}^{\pi}{1-\cos(y)~dy} = \left[ y-\sin(y) \right]_{0}^{\pi} = \pi}\)

\(\pi\)

(15 points) Set up (but do not evaluate) an integral to calculate the volume of the prism in the first octant bounded by the planes \(y=3-3x\) and \(z=2\).

From the equations in the problem statement, we know that the range on z is \(0 \leq z \leq 2\). The ranges on x and y can be derived from the above graph. There are two possible valid orientations, vertical and horizontal.

In this case the ranges on x and y are \(0 \leq x \leq 1\) and \(0 \leq y \leq 3-3x\). This gives these three possible correct answers.

In this case the ranges on x and y are \(0 \leq y \leq 3\) and \(0 \leq x \leq (3-y)/3\). This gives another three possible correct answers.

(15 points) Set up (but do not evaluate) an integral in cylindrical coordinates to calculate the volume of the region bounded by the plane \(z=\sqrt{29}\) and the hyperboloid \(z=\sqrt{4+x^2+y^2}\).

For starters, since we are asked to use cylindrical coordinates, we use the identity \(r^2=x^2+y^2\) to get \(z=\sqrt{4+r^2}\). Plotting the volume where it intersects in the yz-plane (x=0), we can find the z range as \(\sqrt{4+r^2} \leq z \leq \sqrt{29}\). In the xy-plane, we have a circle of radius 5, so that \(0 \leq r \leq 5\). This gives us the following three possible correct answers.

(20 points) Set up (but do not evaluate) an integral in spherical coordinates to calculate the volume of the region inside the cone \(z=(x^2+y^2)^{1/2}\) that lies between the planes \(z=1\) and \(z=2\).

The plot above is the yz-plane trace (the intersection of the solid and the yz-plane). The angle lines are \(y=x\) and \(y=-x\). In spherical coordinates, these lines correlate with the angle \(\phi=\pi/4\). The equations for \(z=1\) and \(z=2\) in spherical coordinates are \(\rho=\sec \phi\) and \(\rho = 2\sec \phi\) giving us the range on \(\phi\) as \(\sec \phi \leq \rho \leq 2\sec \phi\).

The range on \(\theta\) is \(0 \leq \theta \leq 2\pi\) and on \(\phi\) we have \(0 \leq \phi \leq \pi/4\). We can now set up the integral. There are three possible correct answers.