# Calculus 1 : How to find area of a region

## Example Questions

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

What is the average value of the function f(x) = 12x3 + 15x + 5 on the interval [3, 6]?

1350.2

1542

895.67

1302.5

771

1302.5

Explanation:

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: 3x4 + 7.5x2 + 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 #1 : How To Find Area Of A Region

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

2
0
-2
3
-3
Explanation:

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 #1 : How To Find Area Of A Region

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

Explanation:

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:

Find

Explanation:

Find

Explanation:

### Example Question #6 : How To Find Area Of A Region

What is the area of the space below  and above

Explanation:

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 #2 : How To Find Area Of A Region

What is the area below  and above the -axis?

Explanation:

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 #2 : Area

Find the area between the curves  and .

Explanation:

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 #1 : How To Find Area Of A Region

Find the value of

Explanation:

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 #2 : How To Find Area Of A Region

Find the average value of  on the interval