### All Calculus 1 Resources

## Example Questions

### Example Question #1 : How To Find Area Of A Region

What is the average value of the function f(x) = 12x^{3} + 15x + 5 on the interval [3, 6]?

**Possible Answers:**

1350.2

1542

771

895.67

1302.5

**Correct answer:**

1302.5

To find the average value, we must take the integral of f(x) between 3 and 6 and then multiply it by 1/(6 – 3) = 1/3.

The indefinite form of the integral is: 3x^{4} + 7.5x^{2} + 5x

The integral from 3 to 6 is therefore: (3(6)^{4} + 7.5(6)^{2} + 5(6)) - (3(3)^{4} + 7.5(3)^{2} + 5(3)) = (3888 + 270 + 30) – (243 + 22.5 + 15) = 3907.5

The average value is 3907.5/3 = 1302.5

### Example Question #2 : Area

Find the dot product of **a** = <2,2,-1> and **b** = <5,-3,2>.

**Possible Answers:**

**Correct answer:**2

To find the dot product, we multiply the individual corresponding components and add.

Here, the dot product is found by:

2 * 5 + 2 * (-3) + (-1) * 2 = 2.

### Example Question #3 : Area

Find the area of the region enclosed by the parabola and the line .

**Possible Answers:**

**Correct answer:**

The limits of the integration are found by solving and for :

The region runs from to . The limits of the integration are , .

The area between the curves is:

### Example Question #1 : Area

Find

**Possible Answers:**

**Correct answer:**

### Example Question #5 : Area

Find

**Possible Answers:**

**Correct answer:**

### Example Question #6 : Area

What is the area of the space below and above

**Possible Answers:**

**Correct answer:**

is only above over the interval . Areas are given by the definite integral of each function and

The area between the curves is found by subtracting the area between each curve and the -axis from each other. For this area is and for the area is giving an area between curves of

### Example Question #7 : Area

What is the area below and above the -axis?

**Possible Answers:**

**Correct answer:**

To find the area below a curve, you must find the definite integral of the function. In this case the limits of integration are where the original function intercepts the -axis at and . So you must find which is evaluated from to . This gives an answer of

### Example Question #8 : Area

Find the area between the curves and .

**Possible Answers:**

**Correct answer:**

To solve this problem, we first need to find the point where the two equations are equal. Doing this we find that

.

From this, we see that the two graphs are equal at and . We also know that for , is greater than .

So to find the area between these curves we need to evaluate the integral .

The solution to the integral is

.

Evaluating this at and we get

### Example Question #9 : Area

Find the value of

**Possible Answers:**

**Correct answer:**

To solve this problem, we will need to do a -substitution. Letting

.

Substituting our function back into the integral, we get

Evaluating this at and we get

### Example Question #10 : Area

Find the average value of on the interval

**Possible Answers:**

**Correct answer:**

The average is given by integration as:

This means that: