# AP Calculus BC : Application of Integrals

## Example Questions

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### Example Question #1 : Initial Conditions

The temperature of an oven is increasing at a rate  degrees Fahrenheit per miniute for  minutes. The initial temperature of the oven is  degrees Fahrenheit.

What is the temperture of the oven at ? Round your answer to the nearest tenth.

Explanation:

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.

### Example Question #2 : Initial Conditions

Find the equation for the velocity of a particle if the acceleration of the particle is given by:

and the velocity at time  of the particle is

Explanation:

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:

### Example Question #3 : Initial Conditions

Find the work done by gravity exerting an acceleration of   for a   block down  from its original position with no initial velocity.

Remember that

, where  is a force measured in  is work measured in , and  and  are initial and final positions respectively.

Explanation:

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 .

### Example Question #4 : Initial Conditions

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

Explanation:

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:

### Example Question #1 : Volume Of A Solid

Determine the volume of the solid obtained by rotating the region with the following bounds about the x-axis:

Explanation:

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:

### Example Question #2 : Volume Of A Solid

Suppose the functions , and  form a closed region. Rotate this region across the x-axis. What is the volume?

Explanation:

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

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.

### Example Question #3 : Volume Of A Solid

Find the volume of the solid generated by revolving the region bounded by  and the -axis in the first quadrant about the -axis.

Explanation:

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:

### Example Question #4 : Volume Of A Solid

A man fills up a cup of water by leaving it outside during a rainstorm. The rate at which the height of the cup changes is equal to . What is the height of water at ? Assume the cup is empty at .

Explanation:

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.

### Example Question #5 : Volume Of A Solid

Approximate the volume of a solid in the first quadrant revolved about the y-axis and bounded by the functions:   and .  Round the volume to the nearest integer.

Explanation:

Write the washer's method.

Set the equations equal to each other to determine the bounds.

The bounds are from 0 to 3.

Determine the big and small radius.  Rewrite the equations so that they are in terms of y.

Set up the integral and solve for the volume.

The volume to the nearest integer is:

### Example Question #6 : Volume Of A Solid

Determine the volume of a solid created by rotating the curve  and the line  by revolving around the -axis.

Explanation:

Write the volume formula for cylindrical shells.