# AP Calculus AB : Integrals

## Example Questions

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### Example Question #1 : Fundamental Theorem Of Calculus

Evaluate .

Does not exist

Explanation:

Even though an antideritvative of  does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.

. Start

. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , , and then substituting  in the integrand. Lastly the Theorem states you must multiply your result by  (similar to the directions in using the chain rule).

.

### Example Question #2 : Integrals

The graph of a function  is drawn below. Select the best answers to the following:

What is the best interpretation of the function?

Which plot shows the derivative of the function ?

Explanation:

The function  represents the area under the curve  from  to some value of .

Do not be confused by the use of  in the integrand. The reason we use  is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of  with a dummy index  when we write down the integral. It does not change the fundamental behavior of the function  or

The graph of the derivative of  is the same as the graph for . This follows directly from the Second Fundamental Theorem of Calculus.

If the function  is continuous on an interval  containing , then the function defined by:

has for its' derivative

### Example Question #3 : Integrals

Evaluate

Explanation:

Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.

Instead, we make a clever observation of the graph of

Namely, that

This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that

### Example Question #4 : Integrals

A projectile is shot up from a platform  above the ground with a velocity of . Assume that the only force acting on the projectile is gravity that produces a downward acceleration of . Find the velocity as a function of .

Explanation:

with initial conditions

Separate velocity variables and solve.

Plug in intial conditions

### Example Question #5 : Integrals

A gun sends a bullet straight up with a launch velocity of 220 ft/s. It reaches a height of  after  seconds. What is its velocity 500 ft into the air?

Explanation:

The bullet will be at a height of 500 ft on the way up and on the way down.

We use the position equation  to solve for how long it will take to reach a height = 500. and  seconds.

We then plug that into the velocity equation, which is the derivative of the position function. .

We can see that plugging in the value of  yields  and  yields . The positive and negative values of velocity indicates the up and down direction of travel.

### Example Question #6 : Integrals

The position of a particle as a function of time is given below:

At what values of  does the particle change direction?

Explanation:

In order to find the point at which the particle changes direction, we must determine whenever the velocity of the particle changes sign (from positive to negative, or from negative to positive).

We will need to have the function of the particle's velocity before we can determine where the velocity changes sign. Because the velocity is the derivative of position with respect to time, we can write the function for velocity, , as follows:

If we set , then we can determine the points where it can change sign.

The possible points where  will change signs occur at . However, we need to check to make sure.

First, we can try a value less than 2, such as 1, and then a value between 2 and 4, such as 3. We will evaluate  at  and  and see if the sign of the velocity changes.

Thus,  is indeed a point where the velocity changes sign (from positive to negative). This means that the particle does in fact change direction at .

Lastly, we will evaluate the velocity at a value of  larger than 4, such as 5.

The sign of the velocity has switched back to positive, so the particle does indeed change direction at .

### Example Question #7 : Integrals

The speed of a car traveling on the highway is given by the following function of time:

Note that

What does this mean?

The car is not decelerating at time .

The car is not accelerating at time .

The car is not moving at time .

The car's speed is constantly changing at time .

The car takes  seconds to reach its maximum speed.

The car is not moving at time .

Explanation:

The function  gives you the car's speed at time . Therefore, the fact that  means that the car's speed is  at time . This is equivalent to saying that the car is not moving at time . We have to take the derivative of  to make claims about the acceleration.

### Example Question #8 : Integrals

A jogger leaves City  at.  His subsequent position, in feet, is given by the function:

,

where  is the time in minutes.

Find the velocity of the jogger at 15 minutes.

Explanation:

To find velocity, one has to use the first derivative of :

.

Note the units have to be ft/min.

### Example Question #9 : Integrals

Write the equation of a tangent line to the given function at the point.

y = ln(x2) at (e, 3)

y – 3 = (2/e)(x – e)

y = (2/e)(x – e)

y – 3 = (x – e)

y – 3 = ln(e2)(x – e)

y = (2/e)

y – 3 = (2/e)(x – e)

Explanation:

To solve this, first find the derivative of the function (otherwise known as the slope).

y = ln(x2)

y' = (2x/(x2))

Then, to find the slope in respect to the given points (e, 3), plug in e.

y' = (2e)/(e2)

Simplify.

y'=(2/e)

The question asks to find the tangent line to the function at (e, 3), so use the point-slope formula and the points (e, 3).

y – 3 = (2/e)(x – e)

### Example Question #10 : Integrals

Find the equation of the tangent line at  when