# Calculus 1 : Functions

## Example Questions

### Example Question #21 : How To Find Midpoint Riemann Sums

Solve the integral

using Simpson's rule with  subintervals.

Explanation:

Simpson's rule is solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is  therefore

### Example Question #22 : How To Find Midpoint Riemann Sums

Approximate the area under the curve from  to  for the function

using midpoint Riemann sum when .

Explanation:

The find the area under the curve for the function you use the formula .

Riemann's midpoint formula states that

This method breaks the area under the curve into n rectangles. The first term of the equation that is multiplied out find the base length of each rectangle and then the f(x) terms are the heights at the middle of each rectangle. By multiplying the two terms, you find the area and when you add them together you find the approximation of the area under the curve.

We will plug in our values into this formula. a=0, b=4 and n=4. To find the x values to plug into the function we look at the base length of each rectangle.

so there is a rectangle wall every x=1, thus the midpoints will be x1=0.5, x2=1.5, x3=2.5, x4=3.5.

### Example Question #23 : How To Find Midpoint Riemann Sums

Approximate the  using midpoint Riemann sum when .

Explanation:

The find the area under the curve for the function you use the formula .

Riemann's midpoint formula states that

This method breaks the area under the curve into n rectangles. The first term of the equation that is multiplied out find the base length of each rectangle and then the f(x) terms are the heights at the middle of each rectangle. By multiplying the two terms, you find the area and when you add them together you find the approximation of the area under the curve.

We will plug in our values into this formula. a=1, b=3 and n=4. To find the x values to plug into the function we look at the base length of each rectangle.

So there is a rectangle wall every x=0.5 starting at x=1, thus the midpoints will be x1=1.25, x2=1.75, x3=2.25, x4=2.75.

### Example Question #24 : How To Find Midpoint Riemann Sums

Approximate the  using midpoint Riemann sum when .

Explanation:

The find the area under the curve for the function you use the formula .

Riemann's midpoint formula states that

This method breaks the area under the curve into n rectangles. The first term of the equation that is multiplied out find the base length of each rectangle and then the f(x) terms are the heights at the middle of each rectangle. By multiplying the two terms, you find the area and when you add them together you find the approximation of the area under the curve.

We will plug in our values into this formula. a=1, b=9 and n=4. To find the x values to plug into the function we look at the base length of each rectangle.

so there is a rectangle wall every x=2 starting at x=1, thus the midpoints will be x1=2, x2=4, x3=6, x4=8.

### Example Question #25 : How To Find Midpoint Riemann Sums

Approximate the integral  using midpoint Reimann sums and four midpoints.

Explanation:

The form of a Riemann sum follows:

The interval  can be divided into four intervals of length

Each of these intervals in turn have respective midpoints :

Therefore, the approximation of the integral is:

### Example Question #26 : How To Find Midpoint Riemann Sums

Find the midpoint Reimann sum for the integral of the function  over the interval  using five midpoints.

Explanation:

The form of a Riemann sum follows:

Five intervals can be made of length  with midpoints

Thus, the Reimann sum is:

### Example Question #27 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral  using three midpoints.

Explanation:

The form of a Riemann sum follows:

The interval  can be divided into intervals of length  with midpoints .

Therefore, the integral approximation is:

### Example Question #28 : How To Find Midpoint Riemann Sums

Using midpoint Riemann sums, approximate the integral of  over the interval  with four points.

Explanation:

To solve for the midpoint Riemann sum, the function must be evaluated at x=0.5,1.5,2.5, and 3.5.

These must each be multiplied by the segment inteval

.

These are then summed to yield the midpoint Riemann sum approximation:

### Example Question #29 : How To Find Midpoint Riemann Sums

Use the Midpoint Reimann Sum approximation method to approximate the area between  using four subintervals under the curve .

Explanation:

This approximation method has a few steps. First find out how wide your rectangles are. Our interval is [1,3] and we need 4 subintervals.

tells us that each subinterval is 0.5.

Now we need to figure out high each rectangle is. We're going to be finding those y-values at the midpoints of each triangle; namely, at x=1.25, 1.75, 2.25, and 2.75.

We will then plug those x-values into the function given and then multiply by the width of our rectangles (0.5).

Therefore, our equation looks like:

.

This gives us an area of .

### Example Question #30 : How To Find Midpoint Riemann Sums

Use midpoint Riemann sums to approximate the area between the x-axis and the curve   for  to , using three midpoints.