# Calculus 2 : Introduction to Integrals

## Example Questions

### Example Question #12 : Riemann Sums

Given a function , find the Left Riemann Sum of the function on the interval  divided into four sub-intervals.

Explanation:

In order to find the Riemann Sum of a given function, we need to approximate the area under the line or curve resulting from the function using rectangles spaced along equal sub-intervals of a given interval. Since we have an interval  divided into  sub-intervals, we'll be using rectangles with vertices at .

To approximate the area under the curve, we need to find the areas of each rectangle in the sub-intervals. We already know the width or base of each rectangle is   because the rectangles are spaced  units apart. Since we're looking for the Left Riemann Sum, we want to find the heights  of each rectangle by taking the values of each leftmost function value on each sub-interval, as follows:

Putting it all together, the Left Riemann Sum is

.

### Example Question #13 : Riemann Sums

Given a function , find the Left Riemann Sum of the function on the interval  divided into three sub-intervals.

Explanation:

In order to find the Riemann Sum of a given function, we need to approximate the area under the line or curve resulting from the function using rectangles spaced along equal sub-intervals of a given interval. Since we have an interval  divided into  sub-intervals, we'll be using rectangles with vertices at .

To approximate the area under the curve, we need to find the areas of each rectangle in the sub-intervals. We already know the width or base of each rectangle is   because the rectangles are spaced  units apart. Since we're looking for the Left Riemann Sum, we want to find the heights  of each rectangle by taking the values of each leftmost function value on each sub-interval, as follows:

Putting it all together, the Left Riemann Sum is

.

### Example Question #14 : Riemann Sums

Given a function , find the Left Riemann Sum of the function on the interval divided into three sub-intervals.

Explanation:

In order to find the Riemann Sum of a given function, we need to approximate the area under the line or curve resulting from the function using rectangles spaced along equal sub-intervals of a given interval. Since we have an interval divided into sub-intervals, we'll be using rectangles with vertices at .

To approximate the area under the curve, we need to find the areas of each rectangle in the sub-intervals. We already know the width or base of each rectangle is because the rectangles are spaced units apart. Since we're looking for the Left Riemann Sum, we want to find the heights of each rectangle by taking the values of each leftmost function value on each sub-interval, as follows:

Putting it all together, the Left Riemann Sum is

.

### Example Question #15 : Riemann Sums

Given a function , find the Left Riemann Sum of the function on the interval  divided into four sub-intervals.

Explanation:

In order to find the Riemann Sum of a given function, we need to approximate the area under the line or curve resulting from the function using rectangles spaced along equal sub-intervals of a given interval. Since we have an interval  divided into  sub-intervals, we'll be using rectangles with vertices at .

To approximate the area under the curve, we need to find the areas of each rectangle in the sub-intervals. We already know the width or base of each rectangle is  because the rectangles are spaced  unit apart. Since we're looking for the Left Riemann Sum, we want to find the heights  of each rectangle by taking the values of each leftmost function value on each sub-interval, as follows:

Putting it all together, the Left Riemann Sum is

.

### Example Question #16 : Riemann Sums

Given a function , find the Left Riemann Sum of the function on the interval  divided into two sub-intervals.

Explanation:

In order to find the Riemann Sum of a given function, we need to approximate the area under the line or curve resulting from the function using rectangles spaced along equal sub-intervals of a given interval. Since we have an interval  divided into  sub-intervals, we'll be using rectangles with vertices at .

To approximate the area under the curve, we need to find the areas of each rectangle in the sub-intervals. We already know the width or base of each rectangle is  because the rectangles are spaced  units apart. Since we're looking for the Left Riemann Sum, we want to find the heights  of each rectangle by taking the values of each leftmost function value on each sub-interval, as follows:

Putting it all together, the Left Riemann Sum is

.

### Example Question #17 : Riemann Sums

Calculate the value of the Left-Riemann sum of the function

on the interval  divided into five sub-intervals.

Explanation:

The interval  divided into five sub-intervals gives rectangles with vertices of the bases at

For the Left Riemann sum, we need to find the rectangle heights which values come from the left of the sub-intervals, or , and .

Because each sub-interval has a width of , the Left Riemann sum is

### Example Question #18 : Riemann Sums

Evaluate the right Riemann sum of the function

on the interval  divided into three sub-intervals.

Explanation:

The interval  divided into three sub-intervals gives rectangles with vertices of the bases at

For the right Riemann sum, we need to find the rectangle heights which values come from the right of the sub-intervals, or , , and .

Because each sub-interval has width

### Example Question #19 : Riemann Sums

Estimate the area under the curve of f(x)= using Left Riemann sum approximation on the interval from  to  using  rectangles of equal width.

Explanation:

Using the Left Riemann Sum approximation, we first must figure out the width of each rectangle.  Since the problem statement says we have five rectangles, we can use the width equation to determine the width:

, where  stands for width,  are your upper and lower bounds, respectively, and  is the number of rectangles.  The area of a rectangle is given by , so all we need now is to determine the length of each rectangle.  That is found by the value predicted from the original function equation.

Since we are using the left approximation, we need to start on the left most value on each sub-interval, plug that into our function, and use that value as our length.  Then, we simply add up the areas of all five rectangles to get the answer.