In order to evaluate this integral, we will need to use partial fraction decomposition.

Multiply both sides of the equation by the common denominator, which is

This means that  must equal 1, and

The answer is .

To integrate, we must first make the following substitution:

Rewriting the integral in terms of u and integrating, we get

The following rule was used to integrate:

Finally, we replace u with our original x term:

Integrating  over an interval  will tell us the total accumulation, or change, in temperature over that interval. Therefore, we will need to evaluate the integral 

to find the change in temperature that occurs during the first five minutes.

A substitution is useful in this case. Let. We should also express the limits of integration in terms of . When , and when  Making these substitutions leads to the integral

.

To evaluate this, you must know the antiderivative of an exponential function.

In general, 

.

Therefore, 

. 

This tells us that the temperature rose by approximately  degrees during the first five minutes. The last step is to add the initial temperature, which tells us that the temperature at  minutes is 

 degrees.

In order to find the velocity function, we must integrate the accleration function:

We used the rule

to integrate.

Now, we use the initial condition for the velocity function to solve for C. We were told that 

so we plug in zero into the velocity function and solve for C:

C is therefore 30.

Finally, we write out the velocity function, with the integer replacing C:

The force of gravity is proportional to the mass of the object and acceleration of the object. 

Since the block fell down 5 meters, its final position is  and initial position is .

Evaluate the following integral and find the specific function which satisfies the given initial conditions:

To solve this problem,we need to evaluate the given integral, then solve for our constant of integration.

Let's begin by recalling the following integration rules:

Using these two, we can integrate f(x)

SO, we get:

We are almost there, but we need to find c. To do so, plug in our initial conditions and solve:

So, our answer is:

From calculus, we know the volume of an irregular solid can be determined by evaluating the following integral:

Where A(x) is an equation for the cross-sectional area of the solid at any point x. We know our bounds for the integral are x=1 and x=4, as given in the problem, so now all we need is to find the expression A(x) for the area of our solid.

From the given bounds, we know our unrotated region is bounded by the x-axis (y=0) at the bottom, and by the line y=x^2-4x+5 at the top. Because we are rotating about the x-axis, we know that the radius of our solid at any point x is just the distance y=x^2-4x+5. Now that we have a function that describes the radius of the solid at any point x, we can plug the function into the formula for the area of a circle to give us an expression for the cross-sectional area of our solid at any point:

We now have our equation for the cross-sectional area of the solid, which we can integrate from x=1 to x=4 to find its volume:

Write the formula for cylindrical shells, where  is the shell radius and  is the shell height.

Determine the shell radius.

Determine the shell height.  This is done by subtracting the right curve, , with the left curve, .

Find the intersection of  and  to determine the y-bounds of the integral.

The bounds will be from 0 to 2.  Substitute all the givens into the formula and evaluate the integral.

Since we are revolving a region of interest around a horizontal line , we need to express the inner and outer radii in terms of x.

Recall the formula:

The outer radius is  and the inner radius is . The x-limits of the region are between  and . So the volume set-up is:

Using trigonometric identities, we know that:    

Hence:

The rate at which the height changes is , which means .

To find the height after nine seconds, we need to integrate to get .

We can multiply both sides by  to get  and then integrate both sides.

This gives us 

.

Since the cup is empty at , so .

This means . No units were given in the problem, so leaving the answer unitless is acceptable.