#### To determine

**To evaluate:**

The Riemann sum for fx=cosx, 0≤x≤3π/4 with n=6 by taking the sample points to be the left endpoints and illustrate with a diagram what the Riemann sum represents.

#### Answer

a) L6≈1.033186

b) The Riemann sum represents the sum of the areas of the four rectangles above the x-axis minus the sum of the area of rectangle below the x-axis. That is it is the net area of the rectangles with respect to the x-axis. A sixth rectangle is degenerate with height 0 and has no area.

#### Explanation

**1) Concept:**

To evaluate Riemann sum for the left endpoints by using formula of the Riemann sumand draw the figure from the calculated values and explain the behavior of the Riemann sum

**2) Theorem(4):**

Riemann sum formula using left end points is ∑i=1nf(xi-1)∆x iswhere ∆x=b - an, n is the number of the subintervals and xi=a+i∆x

**3) Given:**

fx=cosx, 0≤x≤3π/4

**4) Calculation:**

Here, n=6, a=0 and b=3π/4.

The width of the subintervals is

∆x=b-an

Substitute the values of a and b to get

∆x=3π/4-(0)6

∆x=3π/46=π8

By using the width ∆x=π8 form the subintervals

0,π8, π8, 2π8, 2π8, 3π8, 3π8,4π8, 4π8, 5π8and 5π8, 3π4

The left endpoints are

x0=0, x1=π8, x2=2π8, x3=3π8, x4=4π8, x5=5π8

By using the Riemann sum formula,

L6=∑i=16f(xi-1)∆x

L6=∆x(fx0+fx1+fx2+fx3+fx4+fx5)

Substitute values of the left endpoints and ∆x in above step.

L6=π8(f0+fπ8+f2π8+f3π8+f4π8+f(5π8))

L6=π8(cos(0)+cosπ8+cos2π8+cos3π8+cos4π8+cos5π8)

Simplify right side.

L6≈1.033186

The Riemann sum represents the sum of the areas of the blue rectangles above the x-axis minus the sum of the area of the yellow rectangle below the x-axis. It is the net area of the rectangles with respect to x-axis. A sixth rectangle is degenerate with height 0 and has no area.

**Conclusion:**

a) L6≈1.033186

b) The Riemann sum represents the sum of the areas of the four rectangles above the x-axis minus sum of the area of rectangle below the x-axis. It is the net area of the rectangles with respect to the x-axis. A sixth rectangle is degenerate with height 0 and has no area.