#### To determine

a)

**To find:**

The Riemann sum for the left endpoints with six subintervals

#### Answer

≈8.1

#### Explanation

**1) Concept:**

Definition of the Riemann sum:

∑i=1nfxi* ∆x

is called as the Riemann sum

Where,

∆x= b-an, xi=a+i ∆x, & xi*∈xi-1, xi

Riemann sum is the sum, of the areas of the rectangles that lie above the *x*-axis and the negative of the areas of the rectangles that lie below the *x*-axis

2) **Calculation:**

First, write the expression of the Riemann sum for left endpoints

Six subintervals are given, that means, n=6

By definition of Riemann sum,

∑i=1nfxi ∆x=∑i=16fxi ∆x

So now calculate ∆x

∆x= b-an=6-06=1

Substitute this value in the Riemann sum expression

=∑i=16fxi1

=f0+f1+f2+f3+f4+f5

By using the given graph to approximate the values

≈2+3.6+4+2+-1+-2.5

≈8.1

Draw the diagram of Riemann sum for left endpoints

Riemann sum represents the sum of the areas of the blue rectangles that lie above the *x*-axis minus the sum of the areas of the green rectangles that lie below the *x*-axis.

**Conclusion:**

Therefore, the Riemann sum for left endpoints with six subintervals is

∑i=1nfxi ∆x≈8.1

#### To determine

b)

**To find:**

The Riemann sum for midpoints with six subintervals

#### Answer

≈5.8

#### Explanation

**1) Calculation:**

First write the expression of the Riemann sum for midpoints

Six subintervals are given, that means, n=6

By definition of Riemann sum,

∑i=1nfxi ∆x=∑i=16fxi- ∆x

So now calculate the ∆x,

∆x= b-an=6-06=1

Substitute this value in Riemann sum expression,

=∑i=16fxi-1

=f0.5+f1.5+f2.5+f3.5+f4.5+f5.5

By using the given graph to approximate the values

≈3+3.9+3.5+0.3+-2+-2.9

≈5.8

Draw the diagram of Riemann sum for midpoints,

Riemann sum represents the sum of the areas of the blue rectangles that lie above the *x*-axis minus the sum of the areas of the green rectangles that lie below the *x*-axis.

**Conclusion:**

Therefore, the Riemann sum for midpoints with six subintervals is

∑i=1nfxi ∆x≈5.8