# Calculus 2 : Integrals

## Example Questions

### Example Question #2 : Numerical Approximations To Definite Integrals

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 #3 : Numerical Approximations To Definite Integrals

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 #1 : 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  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 #2 : 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 #3 : 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  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 #1 : Riemann Sum: Right Evaluation

Given a function , find the Right 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 Right Riemann Sum of , we want to find the heights  of each rectangle by taking the values of each rightmost function value on each sub-interval, as follows:

Putting it all together, the Right Riemann Sum is

.

### Example Question #4 : 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 #5 : 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 #6 : 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 #11 : Riemann Sums

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