Need help solving this problem

Question

Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the right-hand endpoints of the subintervals. (See Example 1.)

f(x) = 9 − x² from x = 1 to x = 3; 4 subintervals

Answer 1

The right endpoints fall at x = 1.5, x = 2, x = 2.5 and x = 3. The function values are shown in the picture. The area of each rectangle is its height times its width. The widths are all 0.5.

Area of 1st rectangle: (0.5)(6.75) = 3.375

Area of 2nd rectangle: (0.5)(5) = 2.5

Area of 3rd rectangle: (0.5)(2.75) = 1.375

Area of 1st rectangle: (0.5)(0) = 0

Approx area = 3.375 + 2.5 + 1.375 + 0 = 7.25

Answer 2

f(x) = 9 - x², interval [1,3] with 4 subintervals.

(3-1)/4 = 2/4 = 0.5 so each subinterval is 0.5

x₀ = 1, x₁ = 1.5, x₂ = 2, x₃ = 2.5, x₄ = 3 and Δx = 0.5

So we need to find i = 1→3 ∑f(w_i)Δx_i Δx=0.5

∑f(xi)Δxi = f(1.5)(0.5) + f(2)(0.5) + f(2.5)(0.5) + f(3)(0.5)

= (9-(1.5)²)(0.5) + (9-(2)²)(0.5) +(9-(2.5)²)(0.5) +(9-(3)²)(0.5)

= 3.375 + 2.5 + 1.375 + 0 = 7.25

So the approximation using 4 subintervals is 7.25.

(by the way, the actual area under the graph comes out to be 9 1/3. Considering the approximation on only 4 subintervals this sounds about right.)