Determining the Area Under a Curve Using Rectangles and Integration

Let's calculate the left Riemann sum for n=8; 

Riemann sum using the midpoint of subinterval; 

Where; 

Apply Riemann formula; 

Calculate subintervals; 

Let's calculate the left Riemann sum for n=12; 

Riemann sum using the midpoint of subinterval; 

Where; 

The actual area under the curve using integration is; 

Take the constant out: 

Take the constant out: 

Therefore, 

Apply u-substitution: 

Hence, 

Apply the power rule: 

Therefore, 

Simplify, 

Compute the boundaries: 

The value of the area from integral exceeds the value of the area from rectangles. The difference is due to the area left when calculating the area through the rectangle. Thus, the rectangle method results in the loss of some area that is under the curve. The integration method includes the total exact area under the curve.