#### To determine

**(a)**

**To write:**

An expression for a Riemann sum of a function f on interval [a, b] and explain the meaning of notation

#### Answer

Expression for Riemann sum:Rn=∑i=1nf(xi*)∆x

Where, ∆x is the width of the subinterval and xi* lies in ith subinterval

#### Explanation

**1) Concept:**

The definition of Riemann sum

**2) Calculation:**

If f is a function defined for a≤x≤b, we divide the interval [a, b] into n subintervals of equal width ∆x=b-an then Riemann sum is given by

Rn=∑i=1nf(xi*)∆x

Where, xi* lies in ith subinterval [xi-1, xi]

**Conclusion:**

Therefore,

Expression for Riemann sum:Rn=∑i=1nf(xi*)∆x

Where, ∆x is width of subinterval and xi* lies in ith subinterval

#### To determine

**(b)**

**To explain:**

The geometric interpretation of Riemann sum when fx≥0 with diagram.

#### Answer

The Riemann sum represents the sum of areas of n-rectangles each of whose width is ∆x and the height of ith rectangle is f(xi*). Where f(xi*) and ∆x are defined as above. It is an estimate of area under the curve f(x) and above x-axis.

#### Explanation

**1) Concept:**

Give the geometric interpretation of Riemann sum from the diagram

**2) Calculation:**

fx≥0. It means f is always positive.

Therefore, the Riemann sum can be interpreted as the sum of areas of rectangles

Rn=fx1∆x+fx2∆x+ ………+f(xn)∆x

Where,

∆x=b-an is the width of the rectangle

It is illustrated by the following diagram

From the diagram the Riemann sum represents the sum of areas of blue rectangles.

**Conclusion:**

Therefore,

When fx≥0, the Riemann sum represents the sum of the areas of rectangles.

#### To determine

**(c)**

**To explain:**

The geometric interpretation of the Riemann sum when fx takes both positive and negative values with the help of the diagram

#### Answer

The Riemann sum represents the sum of the areas of rectangles above the x-axis minus the sum of the areas of rectangle below the x-axis that is the net area of the rectangles with respect to x-axis.

#### Explanation

**1) Concept:**

Give the geometric interpretation of Riemann sumfrom the diagram

**2) Calculation:**

When f(x) takes both positive and negative values then,

The Riemann sum represents the sum of the areas of rectangles above the x-axis minus the sum of the areas of rectangle below the x-axis that is the net area of the rectangles with respect to

x-axis.

It is illustrated by the following diagram

From the diagram, the Riemann sum represents the sum of the areas of blue rectangles above the x-axis minus sum of the areas of yellow rectangles below the x-axis which is the net area of the rectangles with respect to x-axis.

**Conclusion:**

When f(x) takes both positive and negative values, theRiemann sum represents the sum of the areas of rectangles above the x-axis minus sum of the areas of the rectangle below the x-axis which is the net area of the rectangles with respect to x-axis.